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TDR-63- 3096, Vol III
RTD
TDR
63-3096
Vol III
DESIGN PROCEDURES FOR SHOCK ISOLATION SYSTEMS
OF UNDERGROUND PROTECTIVE STRUCTURES
Volume III
ionse Spectra of Single-Degree-of-Freedom Elastic and Inelastic Systems
Final Report
June 1964
o
WK9fc\
TECHNICAL DOCUMENTARY REPORT NO. RTD TDR-63-3096, Vol III
■ *' vtsr’JJ
CO
Defense
Research and Technology Division
AIR FORCE WEAPONS LABORATORY
Air Force Systems Command
Kirtland Air Force Base
New Mexico
D DC
This research has been funded by the
Atomic Support Agency under WEB No.
DDC-IRA
13.167.
A
Project No, 1080, Task No. 108005
(Prepared under Contract AF 29(601)-4565 by
Veletsos, A. S. and Newmark, N. M.,
Newmark-Hansen and Associates, Urbana, Illinois.)
DISCLAIMER NOTIC
THIS DOCUMENT IS BEST
QUALITY AVAILABLE. THE COPY
FURNISHED TO DTIC CONTAINED
A SIGNIFICANT NUMBER OF
PAGES WHICH DO NOT
REPRODUCE LEGIBLY.
Research and Technology Division
Air Force Systems Command
AIR FORCE WEAPONS LABORATORY
Kirtland Air Force Base
New Mexico
When Government drawings, specifications, or other data arc used for
any purpose other than in connection with' a definitely related Government
procurement operation, the United States Government thereby incurs no
responsibility nor any obligation whatsoever; and the fact that the Government
may have formulated, furnished, or in any way supplied the said drawings,
specifications, or other data, is not to be regarded by implication or other¬
wise as in any manner licensing the holder or any other person or corporation,
or conveying any rights or permission to manufacture, use, or sell any
patented invention that may in any way be related thereto.
This report is made available for study upon the understanding that the
Government's proprietary interests in and relating thereto shall not be im¬
paired. In case of apparent conflict between the Government's proprietary
interests and those of others, notify the Staff Judge Advocate, Air Force
Systems Command, Andrews AF Base, Washington 25, DC.
This report is published for the exchange and stimulation of ideas; it does
not necessarily express the intent or policy of any higher headquarters.
DDC AVAILABILITY NOTICE
Qualified requesters may obtain copies of this report from DDC.
RTD TDR-63-3096, Vol III
FOREWORD
This report is one of five volumes presenting the results of a series of
studies carried out for the Air Force by General American Transportation
Corporation and Newmark-Hansen Associates. The five volumes comprise
RTD TDR-63-3096 and are organized as follows:
Vol.
I
Structure Interior Motions Due to Air Blast Induced
Ground Shock
Vol.
II
Structure Interior Motions Due to Directly Transmitted
Ground Shock
Vol.
III
Response Spectra of Single-Degree-of-Freedom Elastic
and Inelastic Systems
Vol.
IV
Response Spectra of Two-Degree-of-F reedom Elastic
and Inelastic Systems
Vol.
V
Response Spectra of Multi-Degree-of-Freedom Elastic
Systems
Volumes I and II are authored by General American Transportation
Corporation. Volumes III, IV, and V are authored by Newmark-Hansen and
Associates. Volumes II, IV, and V will be published early in 1965.
Acknowledgment is made to Captain H. Auld, Captain D. H. Merkle,
and Lt J. F. Flory of AFWL for their continued cooperation during the course
of the project.
RTD TDR-63-3096, Vol III
ABSTRACT
A discussion is presented of response spectra for single-degree-of-
freedom systems subjected to different forms of ground excitation.
In the study of elastic systems, the sensitivity of the response to variations
in the detailed characteristics of the input motion is discussed. For each
class of forcing function, simple approximate rules are presented for the
construction of response spectra for undamped systems. Simple rules are
described for the construction of spectra for complex input functions by
compounding the spectra for the "dominant" component pulses of the input
function.
In the studies of inelastic systems, primary attention is given to elasto-
plastic systems and, in an exploratory way, to bilinear systems of the
softening type. Response spectra are presented from which the yield
resistance required to limit the maximum deformation of the system to a
prescribed multiple of its limiting elastic deformation can be determined
directly.
The maximum deformation of an inelastic system is related to that of an
elastic system having the same initial slope in its resistance-deformation
diagram and, for certain conditions, simple design rules are formulated for
the construction of deformation spectra for elastoplastic systems in terms of
the corresponding spectra for the associated elastic systems.
PUBLICATION REVIEW
This report has been reviewed and is approved.
JOHN F. FLORY
2Lt USAF
Project Officer
morn
THOMAS S- EO WRY, JFV
Colonel
USAF
Chief, Civil Engineering Branch
PERRW L. HU IE
Colonel USAF
Chief, Research Division
iii
TABLE OF COHTEHTS
!>i!
1. INTRODUCTION . 1-1
1.1 Objectives of Program . 1-1
1.2 Outline of Studies . . 1-2
1.3 Rotation . . 1-6
1.4 Acknowledgment . 1-9
2. RESPONSE OF SINGLE-DEGREE-OF- FREEDOM ELASTIC SYSTEMS . 2-1
2.1 System Considered . . . 2-1
2.2 Response Quantities of Interest . . . 2-1
2.3 Equations of Motion . . . 2-2
2.4 Analogies Between Response Quantities Corresponding to
Different Forms of Ground Excitation . 2-4
2.3 Response Spectra add" Spectral. Quantities . 2-7
2.6 Ground Motions of Interest . 2-10
£.6,1 General . 2-10
2.6.2 Wave Forms of Primary Component . 2*11
2.7 Deformation Spectra for Undamped Systems Subjected to
Half -Cycle Acceleration Pulses . 2-14
2.7.1 Presentation of Data . 2-14
2.7*2 Discussion of Results . 2-16
a. Low Frequency Systems . 2-10
b. High Frequency Systems . 2-19
c. Maximum Values of A . 2-22
2.7.3 Design Rules . 2-24
2.6 Synthesis of Spectra for a Sequence of Half -Cycle
Acceleration Pulses . 2-23
2.9 Deformation Spectra for Undamped Systems Subjected to
Half-Cycle Velocity Pulses . 2-26
2.9*1 Low Frequency Systems . . 2-26
2.9*2 Presentation and Discussion of Results . 2-29
a. Characteristics of Representative Spectra . . . 2-29
b. Effects of Rise Time and Discontinuities in
Acceleration. . . ...... 2-31
c. Maximum Values of V and A . 2.33
2.9.3 Design Rules . 2-36
2.10 Deformation Spectra for Undamped Systems Subjected to
Half-Cycle Displacement Pulses and Pulses with Partial
Recovery . .
2.10.1 ^«ov frequency Systems . 2-37
2.10.2 Presentation and Discussion of Data ........ 2.39
2.10.3 Dwign Rules . t . 2-42
V-
TABLE OF COHTBRTS (Continued)
2.11 Deformation Spectra for Undamped Systems Subjected to
Full-Cycle Displacement Pulses . . . . . . 2-45
2.11.1 Low Frequency Systems . 2-45
2.11.2 Presentation and Discussion of Data . 2-46
2.11.5 Design Rules . 2-48
2.12 Relationship of Computed Results to Field Test Data . . . 2-46
2.15 Deformation Spectra for Damped Systems . 2-50
2.14 Deformation Spectra for a Combination of Simple Pulses. . 2-55
2.15 Deformation Spectra for Bps tags Sub jetted.; to .Earthquake .
Motions . 2-58
2.15.1 Genersl . 2-58
2.15*2 Presentation of Data . 2-59
2.15.3 Relationship Between Characteristics of Input
Motions and Response Spectra . . 2-62
2.16 Spectra for Other Response Quantities . 2-69
2.16.1 Spectra for Relative Velocity . . 2-$9
2.16.2 Comparison of Pseudo-Velocity and True Relative
Velocity ..... . 2-70
2.16.3 Spectra for Absolute Acceleration . 2-71
2.16.4 Spectra for Absolute Velocity and Absolute
Displacement . . . 2-72
3. RESPONSE 0F I1ELASTIC STSTEMS . 3-1
3*1 General . . . 3-1
3.2 Definitions and Fundamental Relations . 3-2
3.3 Response to Limiting Forms of Ground Excitation . 3-3
3.5*1 Instantaneous Displacement Change . 3-4
3*3*2 Instantaneous Velocity Change . 3-5
3*3*5 Instantaneous Acceleration Change . 3-6
3.3.4 Discussion . 3-7
3.4 Relations Between Response of Elastic and Inelastic
Systems ..... . 5-9
3.5 Deformation Spectra For Klastoplastic Systems ...... 3-13
3.5*1 General . 5-13
3*5*2 Spectra for a Half -Cycle Acceleration Pulse. , . . 3-15
3.5.3 Spectra for Half-Cycle Velocity and Displacement
Pulses . . 5-15
Design Rules . 3-16
Relative Effects of Damping and Inelastic Action . 3-19
3.5.4 Spectra for Multiple-Cycle Velocity Pulses .... 3-19
3*5*5 Spectra for Earthquake Notions . . 3-20
REFERENCES . . 4-1
TABLES . 5-1
vi
TABLE OF CONTEXTS (Continued)
Page
FIGURES . 6-1
APPENDIX A TABULATION OF NUMERICAL DATA . . . . A-l
APPENDIX B EXPRESSIONS FOR RESPONSE OF A S INGLE - DEGREE-OP -
FREEDOM SYSTEM . B-l
DISTRIBUTION C-l
SECTION I
INTRODUCTION
1.1 Objective* of Program
The broad objectives of this program were to develop information
regarding the response of equipment in underground Installations when sub¬
jected to the effects of the ground motions induced by a nuclear detonation,
to evaluate the influence and relative importance of the various factors
affecting the response, and to present simplified design rules far specific
conditions.
If the input motion for a system is prescribed as a function of
time, it is generally recognised that the response of the system can be
computed in a straightforward manner by integration of the governing
differential equations of motion, no matter how complicated the motion or
the system may be. However, such computations are generally time-consuming
and are not very appropriate for purposes of preliminary design. The principal
aim of this study was to establish a body of basic information and simplified
rules which would enable the designer to arrive at a reasonable estimate of
the significant effects of a prescribed motion, and to assess the engineering
significance of the various parameters influencing the response of the system
without the need for elaborate computations. Inasmuch as the detailed
characteristics of the input motion are affected by a large number of uncon¬
trollable factors, a special effort has been made to investigate the
sensitivity of the response to the uncertainties involved in defining the
input data.
The study is based on the concept of the response spectrin, and
covers both elastic and inelastic systems with or without damply. In this
1-1
report, only systems having a single degree of freedom are considered.
However, the Information presented can also he used In conjunction with the
modal method of analysis to evaluate the response of multi -degree -of- freedom
elastic systems for which the natural modes of vibration can be uncoupled.
The study of Inelastic systems is devoted mainly to elastoplastlc systems and,
in an exploratory way, to systems with a bilinear resistance of the softening
type.
There are two important considerations in the design of the
foundation far a piece of equipment. First, the foundation itself must have
sufficient strength to withstand the forces that are developed in it without
failure, and secondly, the accelerations or motions that are transmitted to
the equipment and its parts must not be so severe as to cause damage to it,
or to interfere with its operation. It follows then that response spectra
are needed both for the maximum deformation of the system and for the absolute
displacement, velocity and acceleration of its mass. The latter information
may also be used to define the peak values of the motion experienced by the
base of a light system mounted on a structure that may itself respond under
the action of the shock.
If the piece of equipment is attached to a part of the structure
that experiences essentially the same motion as the base of the structure in
which it is housed, then the equipment may be designed for the shock spectrum
applicable to the input motion. However, if it is mounted on a flexible
element such as a beam or floor, which aay itself respond under the influence
of the shock, then both the intensity end the time-history of the motion at
the bate of the equipment may be significantly different from the original
input notion, and the maxleun response of the system can no longer be
1-2
determined from the shock spectrum corresponding to the base input. It Is
obvious that the extent to which the Input motion Is modified Is a function
both of the characteristics of the structure and of the equipment and of its
mode of attachment to the structure. This problem of the Interaction between
the motions of the equipment and of the supporting structure has been given
some attention under this program, but the results of this effort will be
reported separately. Throughout this report, the characteristics of the motion
at the base of the system under investigation are assumed to be known.
1.2 Outline of Studies
The studies described here can be classified into two groups. The
first group is concerned with the response of elastic systems, with or without
damping, having a single degree of freedom. The input motions considered
included several pulse-type excitations, approximating the primary or main
component of the ground motion associated with a nuclear explosion, and two
strong-motion earthquake records representing examples of extremely coaq>lex
ground motions. The pulse-type of excitations include acceleration functions
composed of from one-half to four cycles of oscillation, with corresponding
displacement functions having from one -quarter of a cycle to one complete
cycle of oscillation. The response quantities studied include the spring
deformation, the relative velocity between the mass and the ground, and the
absolute displacement, absolute velocity, sad absolute acceleration ocf the mass.
The objectives of these studies were:
(a) To assess the sensitivity of the various response quantities to
variations in such parameters as the shape, rise time, and periodicity of the
input function.
(b) To develop siaplifled design rules for the construction of response
spectra for the various response quantities to a greater degree of accuracy
than has been possible previously.
1-5
(c) To formulate procedures for the construction of response spectra
for fairly Involved input functions by synthesising the spectre for a series
of simple component inputs.
(d) To study the effect of viscous damping.
The approach used v&s briefly as follows. First, the response of
undamped systems to pulse -like excitations and to a combination of simple
pulses was investigated for a vide range of the parameters involved, and, on
the basis of the information obtained, simple approximate rules were formulated
for the construction of response spectra for the various response quantities.
Hext, the effect of viscous damping was studied for systems with coefficients
of damping up to 100 percent critical subjected to pulse-like excitations.
Finally, to check the applicability of the approximate rules developed to
inputs of extreme complexity, the response of undamped and daaped systems
subjected to earthquake motions was studied. The earthquake motions were tamed
in preference to ground shock records because they are of greater complexity
than those associated with a nuclear explosion, and consequently provide a
severe test on the adequacy of the approximate rules .
The response spectra far the earthquake motions were evaluated for
a much wider range of natural frequencies than has been customary in previous
studies of earthquake effects, so that these spectra could be correlated with
those corresponding to the sljg>le pulses. It is shown that, even for ground
motions of the complexity of strong motion earthquake records, the response
spectra are similar to those for the simple pulses, and that their salient
features can be estimated with reasonable accuracy from the spectra for the
simple pulses, provided the gross characteristics of the acceleration, velocity
and displacement diagrams of the ground are known.
In the analysis of multi -degree-of -freedom elastic systems by the
use of response spectra, an upper bound to the maximum response can be obtained
by taking the sum of the absolute values of the maximum response in the various
natural modes. This approach overestimates the response. Under some conditions,
a better estimate can be made by taking the square root of the sum of the
squares of the modal responses, but the sign of the error cannot be determined
with this approach. A much lover upper bound may be obtained if the maximum
positive and the maximum negative values of the response of single -degree -of -
freedom systems are known separately, both far the forced-vibration and the
free -vibration eras of the motion. A representative number of such spectra is
Included in this report. In addition to being useful in the analysis of
systems with more than one degree of freedom, these generalised spectra provide
a great deal of insight into the behavior of the system, and enable one to
synthesise the response spectrum corresponding to a sequence or combination of
simple pulses from the spectra applicable to the individual pulses of the
input motion.
The second group of studies performed was concerned mainly with the
response of sizzle-degree -of -freedom elastoplastlc systems having equal yield
levels in the two directions of deformation. Some consideration was also given
to systems with a bilinear resistance of the softening type. For these systems,
only the deformation of the spring was investigated. The parameters
studied Include the characteristics of the input motion, and the natural
frequency, yield point deformation, and damping of the systems. The input
motions considered include several pulae-like excitations and the two strong-
motion earthquake records used in the study of slastlc systems. Furthermore,
as an aid in tha interpr station of the results, the effects of an instantaneous
1-5
displacement change, an instantaneous velocity change, and an instantaneous
acceleration change were studied in detail.
The results are summarized in the form of response spectra from
which the yield resistance required to limit the maximum deformation of the
system to a prescribed multiple of its limiting elastic deformation can be
determined directly. On the basis of the information presented, simple design
rules are formulated under certain conditions for the construction of deformation
spectra for elastoplastic systems in terms of the corresponding spectra for
elastic systems having the same initial slope in their resistance-deformation
diagrams.
All response quantities are presented in dimensionless form, in
terms of the maximum value of the appropriate input motion, so that revised
input data can be treated readily as they became available.
Section 2 deals with the response of elastic systems, and Section 3
with the response of inelastic Bystems. The numerical data used to construct
the response spectra presented in this report are tabulated in Appendix A.
Included in this Appendix is also a brief account of the method of solution
used. Finally, in Appendix B is given a sumsary of expressions for the
computation of various response quantities of single -degree -of- freedom elastic
systems with damping.
1.3 notation
The symbols used are defined where they are first introduced, and
the most important ones are suomarlted here.
A ■ pseudo -acceleration, defined as p^U for an
2
elastic system and as p uy for an inelastic
system
1-6
A
o
c
f
g
k
P
Pd
Q
■ mavlmiim V*lue Of A
■ maximum value of A corresponding to motion
during free vibration
■ Qy/QQ ■ reduction factor; see also Eq. J.l. Its
reciprocal is defined as the overload factor
* undamped natural frequency of system, in cps;
for an inelastic system, it represents the
frequency corresponding to the initial elastic
range of behavior
■> acceleration of gravity
« spring constant
■ mass
b Vk/m > undamped circular natural frequency
* p/iT'p2' - damped circular natural frequency
> spring force
■ absolute maximum value of Q for an Inelastic
system
■ absolute maximum value of Q for an elastic system
* yield value of Q
b undamped natural period of system
a natural period of damped system
> time
■ total duration of a pulse
b durations of an acceleration pulse and a
velocity pulse, respectively; used only when
confusion may arise
- rise time to maximum value of a half-cycle pulse
1-7
t
r
XT,*’
r,v
\,r> \,d
U
u
u
max
ml a
u
rise times for an acceleration, velocity and
displacement half-cycle pulse, respectively;
used only vhen confusion may arise
effective rise time, defined in paragraph
preceding Eq. 2.31
values of tr for an acceleration and velocity
pulse, respectively
time of occurrence of the absolute maximum value
of a response quantity
effective duration of an acceleration, velocity
and displacement half-cycle pulse, respectively
duration of dominant half -cycle pulse in an
input function
values of t^ for an acceleration, velocity and
displacement function, respectively
|u I * absolute maximum value of u without
regards to sign
maximum value of U
x-y * relative displacement between mass and
ground * spring deformation
absolute max j mum deformation of an inelastic
system without regards to Agn
maximum positive value of deformation
maximum negative value of deformation
for an elastic system, the numerically greater
of the values of u _ and u . : in Section J it
max min
is used in lieu of U to denote the absolute
maximum value of u without regards to sign
1-8
u_
'av
o,p
0
* yield point deformation
- relative pseudo-velocity, defined as pU for
elastic systems and as puy for inelastic systems
. maximum value of V
■ absolute displacement of mass
« peak extremum value of x, with its appropriate
sign; in Section 5 it refers to its absolute
value
■ lx I > absolute maximum value of x without
• o
regards to sign
■ displacement of ground
V
« average value of y for a half -cycle pulse
■ residual or final value of y
b absolute maximum value of y without regards to
sign
■ maximum value of y for the primary component of
an earthquake motion
■ c/c _ b fraction of critical coefficient of
cr
damping
u ■ u^/Uy ■ ductility factor
1.4 Acknowledgment
The studies reported herein were made with the assistance of
Dr. C. V. Cbelapati and Messrs. V. H. Walker and J. A. lie to. A portion of
the numerical data presented were obtained by Dr. Cbelapati, as a supplement
to his doctoral dlsseratlon prepared under Dr. Veletsos* direction at the
University of Illinois.
1-9
This contract vas technically Monitored by Capt. Barry Auld and
Capt . Douglas M. Merkle, In succession; their guidance and suggestions in the
course of this study are acknowledged with appreciation.
1-10
SECTION 2
RESPONSE Of SgOLB-DBQREl -OF -FREEDOM ELASTIC SYSTEMS
2.1 8ysten Considered
The system considered consists of a rigid mass, m, connected to a
base by a weightless elastic spring and a dashpot exerting a resisting force
which is proportional to the relative velocity between the mass and the base,
as shown in Fig. 2.1. The spring constant is denoted by k and the coefficient
of viscous damping by c. It is assumed that the mass can move only in the
direction of the spring so that the system has a single degree of freedom.
The base of the system will also be referred to as the ground.
The absolute displacement of the mass is denoted by x, the absolute
displacement of the ground by y, and the relative displacement between the
mass sad the ground, the spring deformation, is denoted by u, l.e.,
u - x - y (2.1)
Both x and y refer to the same inertial frame of reference, and their positive
directions coincide. Tbs quantity u Is taken as positive when It produces
tension in the spring. For a fixed-base system acted upon by an exciting
force, the spring deformation, which is also equal to the absolute displace¬
ment of the mass, is designated by x. A dot superscript denotes differentia¬
tion with respect to time. For example, & denotes the relative velocity between
the mass and the ground, and 2 denotes the absolute acceleration of the mass.
2.2 Response Quantities of Interest
Throughout this report, the tern response is used in s generalised
•ease to include war response quantity, such as a force, stress, displacement
or velocity. In a similar manner, the ten disturbance may refer to a loidigg.
2-1
such as a force or pressure, or to a ground notion, which say be described as
a tine function of acceleration, Telocity or displacement. Whatever Its fora,
the forcing function le presumed to be known and Independent of the notion of
the system Itself.
When the source of excitation Is a force, the response quantities of
Interest are the displacement, velocity and acceleration of the mass. Tear a
ground excitation, both the absolute and the relative values of these quanti¬
ties nay be needed. Of greatest Importance is the relative displacement between
the aass and the ground, which is proportional to the force or stress la the
responding structure. The relative velocity, which Is proportional to the rate
of straining of the material of the spring. Is also of Interest, as It may be
used to estimate the possible increase in the yield level of the material under
dynamic conditions and to determine the magnitude of the asxlaias force due to
viscous dating. The relative acceleration between the aass and the ground,
although it does not appear to have any special practical significance. Is
of Interest, because It may be used la conjunction with certain analogies to
obtain response quantities for modified farms of ground excitation. This matter
is discussed further In Section 2 ,k. The absolute displacement, velocity and
acceleration of the aass are needed because the design of the system may be
governed by limitations on the motion of its aasa rather than by strength con¬
siderations, and because these quantities may be uaed to define the character¬
istics of the input motion for a secondary light aaas that may be a part of the
main aass or aay actually be attached to It.
2.3 Equations of Motion
Tor use In subsequent developments, it Is desirable to record here
the governing differential equation of motion. For a system subjected to a
ground excitation, this equation Is
2-2
vher* p(»c/c ) da notes the fraction of critical coefficient of damping, and
p(« yW/a) denotes the undamped circular natural frequency of the system. The
natural frequency of the system in cycles per second is denoted by f , and is
given by the expression
f
Equation 2.2 can be written in one of the following alternate forms:
*i + 2pp i + p2x - p2y(t) + 20p y(t) (2.3)
or
h* + 2pp u + p2u ■ - y(t) (2.V)
The latter form is the more convenient of the two when the ground motion is
specified am am acceleration function. Obviously, the solution of these equa¬
tions depends on the characteristics of the disturbing function, the degree of
damping In the system, as represented by the parameter (3, and the natural fre¬
quency of the system. Actually, the latter parameter enters In the solution
am a dimensionless product of f and a characteristic time of the disturbing
function. The expressions for the various response quantities are given In
Appendix B In terns of Duhamel's integral.
For a fixed-base system acted upon by an erfr-VniiL fore® P(t) applied
at the mass, the governing differential equation Is
5 + 2pp k + p2x - p\t(t) (2.5)
where
(2.6)
denotes the deflection that would be produced by the force P( t) were to be
applied gradually to the system. This quantity will be referred to as the
static deflection of the system, and Its marl sms value will be designated as
Equation 2.5 la analogous to 2q. 2.4, and Its solution nay bs
obtained f res that of Eq. 2 .4 simply by replacing u by z and the quantity
y<t) by sinus p2x>t(t) . it follows that, if the external force P(t) has the
sane shape as the acceleration function for the ground input problem and if,
in addition, the initial conditions on x and i for the force input are the
sane as those on u and u, for the acceleration input, then the amplification
factors x(t)/(xrt)0 and [-p^u(t)]/?0 for the two cases will be identical.
Similar analogies also exist between the derivatives of these quantities. In
particular,
[pTxffi- *“ *° r(t)] " [-**$^ **• *<*>] (2-7)
and
If* ■■■ due to P(t)j - 3^1 doe to y(t)j (2SB)
p ^xst^o °
Thus, if the response histories for one set of quantities, say x,i and 2, are
available, the histories for the corresponding set, u, 4 and 4, can be obtained
directly. Obviously, these analogies are also applicable to the maxi ana values
of the response quantities.
Par a system that le initially at rest, the initial conditions for
the two problems considered above will be the sane if
y(0) - t(0) . 0
2.4 Between Response Quantities Corresponding to Different Pome
of Ground Excitation
Let y^(t), y2(t) end y^(t) be three different notions, such that the
displacement history of the first, the velocity history of the second, and the
acceleration history of the third have the same shape. That is,
TVW *,(») »,(»)
(*.»
where the subscript o denotes the nwirlmai value of the function to which it is
attached. Also, let Xj(t) he the absolute displacement of the mass of the system
subjected to y^(t).
The equations of motion for x^, x^, and x^ can he expressed in terms
of jrp f2 and as follows:
*»i 2
2»r + pxi
- p2yL(t) + 2pp fx( t)
2BP ^ + P2*2
P2tg(t) ♦ 2fip t2(t)
(2.10a)
(2.10b)
♦ »V A
3(t) + tepy5(t)
(2.10c)
the last two equations being obtained formally from Kq. 2.3 hy differentiation.
tor, if the initial conditions for these equations are the same, the three solu¬
tions will he identical, aad it may be concluded that:
[tS4-* '!<*>] ' [
rV*>
due to t2(t)J - due to y?(t)j
(2.U)
By subtracting from the three parts of this equation, the corresponding parts of
Iq. 2.9, aad recalling that T xj ” 7y 000 concludes further that
[tS *“ yi(t)] ’ [w; *• * V*>] ■ •" * yj(t)]
(*.12)
Similar analogies also exist between the hltfier derivatives of these quantities.
The initial conditionsoof Bqs. 2.10 are specified la terms of
x^(0) and Xq,(0) for Bq. 2.10n
i^(0) aad ^(0) for Iq. 2.10b, aad
^(0) and^(O) for Bq. 2.10e.
2-5
The last three quantities can be related to the initial values of the input
notion by application of Eq. 2.2 as follotrs:
*ig(0) - - p^O) - 2ftp yo)
(2.13)
yo) - - p2Uj(o) - 2fip yo)
(2.1k)
yo) . 20p^Uj(o) - (i-Jtf2)p2yo) + 20p yo)
(2.15)
As an illustration, consider the special case of a systea that is initially at
rest. If the initial values of y^, y^ and are xero, the initial conditions
for each of Bqs. 2.10 are likewise xero, and the analogies of Eqs. 2.11 and
2.12 axe valid. On the other hand, if the initial values of y^, and are
different from xero, it can readily be verified that only the initial condi¬
tions for Eqs. 2.10b and 2.10c are Identical, with the result that in Bqs. 2.11
and 2.12 only the analogies represented by the equality of the second and third
terns are valid.
For the special case of a systea without dating, it follows from
Eq. 2.2 that
’i(t) - - p2u(t) (2.16)
and the analogies described in Bqs. 2.11 and 2.12 can therefore be extended
accordingly.
Of special Interest is the following set of analogies applicable to
the free vibration era of the notion for systems without damping, If the terminal
value of t) in Eq. 2.9 is zero, the marl—ia values of the velocity. aM, thg
displacement during free vibration are related by the equations
t) " ■ PV*)
2-6
since the notion Is of the simple hsmonlc type. It follows then that the maximum
values of
Mt) -j rpu«(t)
jjjj; 4u* to yi(t)J - Ltw du* 10
■ [-
p2u_(t)
T97 due 10 Vt}
(2.17)
2.5 Besponse Spectra and Spectral Quantities
For design purposes, it Is generally necessary to know both the maxi-
wsb positive and the maximum negative values of the response. In certain appli¬
cations, It nay also he desirable to know the Magnitudes of these quantities
separately both for the interval that the forcing function acts on the system
and for the tine following the end of the disturbance. If the direction of the
excitation cannot be predicted, or If the characteristics of the exciting func¬
tion are such that the positive and the Marianas negative values of the
response are equally likely, then it nay suffice to know only the absolute sexi¬
sm value of the response.
The subscript "sax" will be used to designate the absolute sari sain
positive value of a response quantity, and the subscript "min" will refer
to the corresponding maximum negative value. Thus
■ the absolute maximal positive deformation
* the absolute maximum negative value of the absolute displacement
The numerically greater of the maxi, mm positive and the maxi mm negative response
quantities will be Identified with the subscript o, and the absolute maxi mm
value of the quantity, without regards to sign, will be denoted by the coital
letter of the syri>ol used to designate that quantity. Thus
0 - |«0I
(2.18a)
D . |40l
(2.18b)
* - l*0l
(2.18c)
2-7
A plot of the maximum value of a response quantity as a function of
the natural frequency of the system, Or a quantity which Is related to the
frequency, constitutes the response spectrum or shock spectrum for that quantity.
It Is assumed that the system has a single degree of freedom, and that the excita¬
tion Is known and Independent of the motion of the system Itself. For exaig>le,
the diagram expressing the variation of with frequency represents the
response spectra* for the absolute maximum positive value of the absolute velocity
of the mass of the system.
It Is convenient to express the various response quantities In dimen¬
sionless form by normalizing them with respect to the maximum value of the
corresponding Input quantity. For a ground motion, displacements may conveni¬
ently be expressed In terms of the ground displacement, velocities In
terms of the maximum ground velocity, etc. The ratio of the Instantaneous
value of a response quantity to the corresponding maximum Input value will be
referred to as the amplification factor for that quantity. The normalized
spectral quantities are the peak values of the amplification factors.
The term deformation spectrum will be used to designate the response
spectrum for the sb solute msxlmum spring deformation, 0, or a quantity used
as a measure of U.
In many Instances, the maximum spring deformation may be expressed
more conveniently by the quantity Y, defined as
▼ - PO (2.19)
where p is the undated circular natural frequency of the system. The quantity
V has units of velocity, end Is related to the maxlmas strain energy of the
system, by the equation
8 _ - £ mY2 (2.20)
2-8
which follows from the fact that
E_
max
2 “P
w
Under certain conditions to be discussed subsequently, the quantity
V is identical to, or approximately equal to, the maxi mum relative velocity,
U, and these quantities have, at times, been used interchangeably. However,
they are generally different from one (mother, and care should be exercised
in replacing one for the other. To avoid possible confusion, the quantity ▼
will be referred to as the relative pseudo-velocity, or simply pseudo* velocity,
and the term relative velocity will be reserved for the true relative velocity
of the system.
Another convenient measure of the maximum spring deformation is the
pseudo-acceleration of the mass A, defined as
A ■ pV ■ p^U (2.21)
and related to the maximum spring force, Qq, as follows:
where V is the weight of the system, and g is the gravitational acceleration.
The force any also be written in the form
Q0-CW (2.22)
where C, the so-called lateral force coefficient or dynamic load factor, repre¬
sents the number of times the system must be capable of supporting its own weight
in the direction of motion, and is equal to the pseudo-ecchleratloa of the system
expressed in units of gravity.
2-9
2
Far a system without damping, the acceleration 'i * - p u, whence it
follows that the quantity A alto represents the absolute maximum value of the
true acceleration of the mass, X. For a damped system, A is only approximately
equal to X, but the difference between these two quantities is of practical
significance only for large values of damping, as will be seen subsequently.
It may finally be noted from Eq. 2.2 that, for a damped system, the value of
2
p u at the Instant that u is an extremum represents the true acceleration of
the mass, since the second term in this equation vanishes by virtue of the fact
that u « 0 at that Instant. It is to be emphasized, however, that the maximum
values of these two quantities are equal only for 0*0.
2.6 Ground Motions of Interest
2.6.1 General. Whereas the detailed characteristics of the ground
motions resulting from two nuclear explosions under comparable conditions may
differ significantly because of unavoidable differences in the values of the
physical parameters Involved, the gross or smoothed-out characteristics of such
motions ere generally quite similar. These similarities are particularly
noticeable in the records of ground velocity end ground displacement.
Examination of available field teat data (Ref. 1)* reveals that the
time -hi story of the ground velocity induced by a nuclear explosion is charac¬
terized by a low-frequency, pulse-type of disturbance on which ere superimposed
oscillations of higher frequencies sad usually smaller applitudes of more or
less random character. The pulse-like disturbance will be referred to as the
primary component of the motion, and the oscillatory component as the secondary
or random component. The general shape, the peek value, end the duration of the
primary component can generally be estimated with fair accuracy in terms
♦Listed at the end of the text.
2-10
of the yield of the weapon, the distance of the point of observation from the
source of the explosion, and the direction of the motion. In contrast, the
random component cannot be defined reliably. This component arises mainly
from reflections of the transmitted shock wave and is influenced significantly
by the detailed properties of the medium through which the shock is transmitted.
Since the properties of the soil may vary in a more or less arbitrary manner
with depth or with distance from ground zero, the characteristics of this
coaqponent can at best be described in statistical terms. It can generally be
said, however, that the less uniform the soil conditions, or the greater the
ground range, the more prominent is the contribution of the random component
to the total input motion.
It is convenient to consider the effects of the two components of
the input motion separately, and to estimate the maximum effect of the actual
input by a combination of the corresponding effects produced by the two com¬
ponent Inputs. The greater part of this report is concerned with the effect
of the primary component of the motion. However, the manner in which the random
component may modify the effects produced by the primary component, may also
be estimated from the data to be presented.
2.6.2 Wave Forms of Primary Component. In the irmiediate vicinity
of ground zero and at shallow depths, the velocity of the ground in the vertical
direction has the characteristic shape of the overpr' .ore curve, as shown in
the lower part of Fig. 2.2a. This is essentially a half -cycle pulse with a
sharp rise to a maximum value followed by a gentler decay. The corresponding
displacement-time curve, shown in the upper part of the figure, is a pulse
with a quarter of a cycle and a final or permanent displacement equal to the
maximal value of the ground displacement.
2-11
As the distance from ground zero Increases, the primary component
of the ground velocity changes Into a more nearly full -cycle pulse with both
positive (downward) and negative (upward) parts. At the shorter ranges, the
area under the negative part of the velocity diagram is smaller than under the
positive part, and, consequently, the displacement-time diagram shows only
partial recovery from Its maximum value, as Indicated In Pig. 2.2b. At the
greater ranges, the two areas become equal to each other, and the ground dis¬
placement Is represented by a half -cycle pulse with complete recovery, as shown
In Fig. 2.2c. In general, the duration of the negative phase of the velocity
pulse Is longer than of the positive phase, and the corresponding displacement
pulse Is very similar to the velocity pulse applicable la the Immediate vicinity
of ground zero. At still greater ranges, the velocity ^diagram nay consist of three
ox* more half cycles., and the asdoelated displacement ilagrammiy ha-ge either one
complete cycle, as shown In Fig. 2. 2d, or, several half-cycle* as discussed in Ref. 2*
Half -cycle displacement pulses with complete recovery may also be
expected In the Immediate vicinity of ground zero If the Intensity of the
shock or the strength of the ground material are such that no permanent die-
placement results.
Evidently, the Intensity of the ground motion decreases with Increas¬
ing ground range, but this reduction In Intensity may not be sufficiently great
to cospensate for the increased dynamic effects resulting from the greater
mmtoer of oscillations present In the input function.
The characteristics of the ground motion in the horizontal direction
are generally similar to those for vertical motion at great ranges. The time-
history of the displacement Is represented either by a half -cycle pulse with
collets recovery, or a pulse with both positive and negative parts. It
should be noted, however, that the available data fear this case are not as
conclusive as those for motion in the vertical direction.
2-12
In sumsmry then, the following forms of ground motion are of interest.
(1) Half -cycle velocity pulses,
(2) Half -cycle displacement pulses, or displacement pulses with
partial recovery from their maximum value.
(3) Tull -cycle displacement pulses coapbsad of both positive and
negative parts, and displacement functions with several half-
cycles .
Inasmuch as it is physically impossible to have instantaneous changes of displace¬
ment, velocity or acceleration of the ground, the displacement-time diagrams
and their first and second derivatives must be continuous functions. In the
following discussion, primary emphasis is given to the effects of continuous
functions as indicated shove; however, some discontinuous pulses are also con¬
sidered as limiting forms of ground excitation. In addition, for the sake of
completeness and for the purpose of developing the various concepts in an
orderly fashion, consideration is first given to the effects of motions repre¬
sented by a half -cycle acceleration pulse. For this class of excitation, the
velocity of the ground after termination of the pulse has a constant value
different from zero, and the ground displacement Increases linearly, as shown
in Fig. 2.3. This type of motion is of course of Interest In the design of
equipment mounted In a moving vehicle.
It oust be noted here that the characteristics of the ground motions
Induced by a nuclear blast are not unlike those obtained for some strong motion
earthquakes, and that,, when properly Interpreted, the dynamic response of
systems to the two sources of excitation is generally quite similar. The earth¬
quake motion is of course of longer duration than the blast Induced notion,
and the random cosqponent of the motion is more pronounced for an earthquake
record. However, insofar as their effect* on systemrwfcth moderate amounts of
daqplogam concerned, these differences are found to be of minor consequence.
2-13
2.7 Deformation Spectra for Undamped Systems Subjected to Half -Cycle
Acceleration Pulses
2.7.I Presentation of Data. In Pigs. 2.4 and 2.5 are given
response spectra for the maximum positive and the maximum negative accelera¬
tion of the mass of a system subjected to a ground acceleration in the form .
of a half -sine pulse and a versed-sine pulse, respectively. The system is
considered to be initially at rest. In each figure, the response acceleration,
*£, normalized with respect to the maximal input acceleration, yQ, is plotted
against the dimensionless product of the natural frequency of the system, in
cycles per second, f , and the duration of the pulse, t^. The results for
the forced- vibration era of the motion, i.e. the period during which the pulse
acts on the system, are given separately from those applicable to the period
of free vibration. In particular, the solid line represent# the spectrum for
the — Tfit— positive acceleration during forced vibration, the dashed-dotted
line represents the spectrum for the corresponding maxi man negative accelera¬
tion, and the dashed line represents the spectrum for the maximum acceleration
during free vibration. In the latter case, the positive and negative values
of the response are numerically equal. Since the system has no dasqplng, by
virtue of Eq. 2.16, these spectra can also be interpreted as deformation spectra.
In Figs. 2.6 and 2.7, the absolute maxi aim values of the positive and
the negative values of the response acceleration, without regards as to their
times of occurreuce, are replotted on logarithmic scales. On such a plot,
diagonal lines sloping upward to the right are lines of constant values of the
ratio of the quantities plotted on the ordinate and the abscissa, and diagonal
lines sloping downward to the right are lines of constant values of the product
of these two quantities. The diagonal scales in these figures have been normalised
such that they represent the dimensionless ratios
2-14
-*S and M '
respectively, where *yQ is the maximum value of the derivative of the input
acceleration function, the so-called "Jerk”.
For input acceleration functions having more than a single half -cycle,
such as those to he considered subsequently, it is convenient to plot the
relative pseudo- velocity pu on the vertical axis of the diagram instead of
on the diagonal axis as was done in Figs. 2.6 and 2.7, and for the sake of
uniformity, the quantity pu will be plotted on the vertical axis for all
deformation spectra given in the remainder of this report.
In Fig. 2.8 the upper envelope of the spectra presented in Fig. 2.6
is replotted in this manner with the pseudo-velocity V normalized by the
maximum ground velocity yQ. The corresponding spectrum for the versed sine
pulse is given in Fig. 2,9 along with those for two "skewed versed sine" pulses
having rise times, tr, equal to l/4 and l/8 the pulse duration. The latter
pulses consist of two half -segments of a versed sine with unequal lengths.
For t J tj « l/4, the duration of the second segment is three times as long as
that of the first. Included in Fig. 2.9, is also a sketch of the derivative
of the input acceleration function.
On a plot such as that given in Fig. 2.8 diagonal lines sloping upward
to the right are lines of constant displacement U, and diagonal lines sloping
downward to the right are lines of constant acceleration, A « X. Accordingly,
with the scales for the diagonal axes established, from a plot of V alone, one
can also read the values of U and A. In Figs. 2.8 and 2.9 the scales for A
have been normalized with respect to the maximum value of the input acceleration
VQ. The relationship between V/yQ and A/V0 nay be stated as
V 1 *0 A
T0 " s*f T0 T0
(2.23)
2-15
*o-fVd
For a half -cycle acceleration pu
and Kq. 2.23 reduce* to
£
~*ir0
Tor a versed-sine pulse of arbitrary rise -duration ratio.
U ■ 5 Vd
and
▼ 1 A:
For the class of Input functions considered In this section, the scale for
the relative displacement Q cannot he normalized with respect to the uaxlmnsi
ground displacement, since this displacement is not defined In this case.
For use In later sections. It Is noted that, when jQ is defined, the relation¬
ship between f/fQ and U/yQ may be stated as
▼. 2* f ^ ^ (2.2b)
Jo Jo Jo
2.7.2 Discussion of Results. From the Information that has been
presented and from a study of the additional data susmarlsed In Keferences 3
through 7, the following Observations can be made.
a. Low Frequency 8ystem». This term describes the condition In
which the duration of the excitation Is small relative to the natural period
of the system, l.e. t^f Is small. From Figs. 2.b through 2.7 it can be seen
that for values of t^f less than about 0.6, the absolute value of the
deformation occurs during free vibration, with the result that both and
are mmmrlcally the seme. As t^f approaches aero, the curves approach
the limiting values at
2-16
1
(2.25a)
and
Vmin
-1
(2.25b)
that this result is as It should be may be appreciated physically by noting
that, with t^f approaching zero, the disturbing function approaches a velocity
step of Infinite duration, for which It Is veil known that the marina and
minimum values of the deformation are as given by Eqs. 2.25. For use subse¬
quently, this result is derived below by application of the simple Impulse
theory.
Consider first a half -cycle force pulse applied to the mass of a
fixed-base system. For small values of t&f , the pulse may be approximated
by an Instantaneous velocity change of the mass, vq, the magnitude of which
may be obtained by application of the ljyulse momentum relation
0
The displacement x may then be determined from the expression*
which yields
v
*(t) = y sin pt
c(t) *p[/x,t(x) dtj sin pt
(2.26)
(2.2?o)
(2.27b)
The ™ value of this expression constitutes an upper bound to the true
displacement, since the effectiveness of the Impulse has been over¬
estimated by assuming It to be concentrated at t - 0 instead of being spread
over a finite time.
"Unless otherwise noted, the limits of integration for the Integral expressions
presented are from 0 to td, and the resulting equations are applicable for
values of t > t^.
2-17
If the disturbing function is a ground acceleration ?(t) of tbs
sane shape as P(t), the resulting spring deformation a my be obtained from
Eqs . 2.26 and 2.27a b y replacing P(t)by t) and x(t) by u(t). The result¬
ing expression is
U(t) . - i [ffM dtj sin pt ■ - ^ y o sin pt (2.28a)
from which Bqs. 2.25 follow directly. In particular,
U-P /V(t) dT "P*6
Vote that u^q occurs at f/k and \iWjt at 5*A> ****** T is the natural period
of the system.
From the available data, the error Incurred by the use of this staple
Impulse theory is estimated to be less than 10 percent if
where t is the effective duration of the acceleration pulse. This quantity
o,a — -
is defined as the duration of a triangular pulse having the same peak value
and the same area as the actual pulse, and it is given by the expression
* _ « jo „ *av .
V r rr
(2.*9)
in which ?av is the average value of the input acceleration. For the versed
sine pulses considered, the effective duration t - t.. The concept of an
effective pulse duration is Introduced to account for circumstances in which
the exciting function Includes low-intensity regions, which, on purely physical
grounds, can be expected to be relatively ineffective. From Eq. 2.25 it can
be verified that the limiting value of t ^f - l/s referred to above, corresponds
to the value of t^f for which both V/yo and a/9q are equal to one. In general,
the effect of pulse shape Is unimportant for values of t f as high as 0.5 •
o,a
b. High Frequency Systems. Pigs. 2.k through 2.J show that for values
of t^f greater than that corresponding to the peak value of the curve for free
vibration, the maximum positive value of x is greater than the maximum negative
value, the difference becoming progressively greater with increasing value of
t^f . As t^f approaches infinity, *£|iu(t and approach the maximum positive
and the maximum negative value of the input function, respectively. For the
class of pulses considered in this section, the negative value is of course
zero. These limiting values are valid only if the input acceleration is a
continuous function.
In general, x(t) can be expressed as the sum of two components: a
function that is proportional to the input acceleration, and a sinusoidal com¬
ponent, the frequency of which is equal to the natural frequency of the system,
f . As f tends to infinity, the magnitude of the first component becomes numeri¬
cally equal to the input acceleration, and, in the absence of any discontinuities
in the input acceleration, the asplltude of the periodic ccaqponent reduces to
zero. The response of the system then approaches that obtained under "static"
conditions, and the limiting value of K becomes equal to the — input
acceleration.
The effect of a discontinuity in the input acceleration is to make
the amplitude of the periodic component in the expression for *i( t) equal to the
magnitude of the discontinuity. If the input function has several discontinui¬
ties, the amplitude of the periodic component at any instant la equal to the
numerical sum of the discontinuities up to that Instant. This condition is
illustrated in Fig. 2.10 for a series of acceleration pulses, including two
full-cycle functions. The dashed line curves in this figure represent the input
2-19
acceleration and the solid lines the acceleration of the system. The curves
are dravn on the assumption that the natural period of the system, as repre¬
sented by the period of the oscillatory coaponent of the response. Is —
In comparison to the smallest time Interval between consecutive discontinuities.
These plots show that the limiting value of A moat be either equal to the sum
of the absolute maximum value of the input acceleration and the numerical sun
of the discontinuities preceding this maximum, or equal to the numerical sum
of a greater number of discontinuities and the magnitude of the following maxi¬
mum, whichever combination gives the numerically greater value. Far example,
for the Input function considered In Fig. 2.10f, the limiting value of A is
*(*o>l * *1* ^ l*i 4 *2 + w tz#ro ♦ \ + *3!
whichever Is greatest. From these plots, the limiting values of
can also be determined, as shown.
It must be remembered that la the preceding discussion, the system
was presumed to be completely undamped. Obviously, the effect of damping Is to
reduce the amplitude of the periodic component of the motion, the magnitude of
the reduction being more pronounced in cases such as those shown In Figs. 2.10c
and 2.1Qf , Where the maximum response occurs at a considerable distance from
the major discontinuity. Instead of when the maximum occurs Immediately after
the discontinuity.
For am Input acceleration pulse without any discontinuities, the range
of frequencies within which the quantity A nay be considered to be equal to the
maximum Input acceleration depends on the shortest rise time of the pulse rather
than on its duration. From available data, and particularly those given In
Fig. t.19 of Ref. 3, It la concluded that the peak value of the Input and the
response accelerations nay be considered to be equal for values of
2-80
(2.30)
l f > 1.25
—
where t Is the shortest "effective" rise time to the peak acceleration. This
r,a
quantity Is defined as the horizontal projection of a straight line extending
from zero to the maxi mom value of the Input acceleration with a slope equal to
the Maximum slope of the original curve, and Is given by the equation
*
r,a
(2.31)
The error incurred by the use of the approximation referred to in Bq. 2.30 Is
estimated to be less than about 15 percent.
The general procedure described In the preceding paragraphs for the
computation of the response acceleration of high-frequency systems, in combina¬
tion with the analogies presented In Art. 2.k, can also be used to define the
limiting values of other response quantities and to obtain other useful Infor¬
mation. For example, the limiting velum of S(t) may be determined by considering
*f(t) to be the associated Input function. Now, If y(t) is a discontinuous
function, the amplitude of x(t) during free vibration will be different from
zero, and from the magnitude of this amplitude, it Is also possible to define
the manner in which approaches Its limiting value. For the half -cycle
acceleration pulses considered In this section, occurs during free vibra¬
tion, and la therefore, related to by the equation
Vu-*3-«
As an Illustration, consider the spectra for the half -cycle acceleration pulse
presented In Fig. 2.6. In this case, 7(t) is a cosine function, and the limit¬
ing value of ^ - - 2fo, as may be appreciated from the diagram la Fig. 2.101.
It follows that at the limit
2-21
this result being substantiated by the data in Fig. 2.6.
For the versed sine acceleration pulse considered in Fig. 2.7, the
Uniting amplitude of *f(t) during free vibration is zero since 'f(t) is a continu¬
ous function. However, by working with the second derivatlee of V( t ) , Which is
discontinuous, and its associated response quantity *x(t) , one finds that the
limiting value of the residual amplitude of *JT(t) is 2
Accordingly,
and by noting that
one obtains the result
which agrees with the data presented in Fig. 2.7.
c. Maximum Values of A. In Table 1 are listed the values of kQ
and Kp with the associated values of t^f for the pulses considered in the preced¬
ing sections and for three triangular pulses discussed in Ref. 5 • The quantity
A denotes the absolute aarlnua value of A, and A_ denotes the wart nun corres-
o r
ponding to the residual or free-vlbration notion. The rise tines of these
pulses, tr, are also listed along with the effective rise tines, as defined
by Hq. 2.J1.
In Table 1 the snallest value of A0/9Q - 1.26 is obtained for a
triangular pulse with vertical termination, sad the greatest value of 2 is
obtained for pulses with a vertical front. The results show clearly that the
2-22
rise time of the pulse is the most important single parameter influencing the
magnitude of the absolute maximum response, the detailed shape of the pulse
being of secondary significance. Between pulses having the sane peak value
and the same duration, the greater value of A^y^ can generally be expected
to occur for the pulse with the shorter effective rise time.
For pulses with a smooth rise, the value of Ar is equal to or slightly
less than the absolute maximum value A . However, the difference between the
o
two sets of values Increases with decreasing rise time, with the maximum differ*
ence obtained for a pulse with vertical front and a smooth decay.
The value of t^f corresponding to A Q/VQ generally increases with
decreasing rise time. Although there does not appear to exist a simple way of
defining this value, it is worth noting that it is consistently greater than
or equal to the value corresponding to the peak residual acceleration, Ar, the
difference between the two values becoming greatest for the pulses with a
sharp rise. The value of t^t corresponding to Ar can most reliably be approxi¬
mated In terms of the effective duration of the pulse, t , as follows
•*0
to f S 0.8 or tdf S 0.4 (2.32)
* ^a v
For an acceleration pulse with a vertical front and a smooth decay,
the response spectrum for k/fQ increases monotonlcally with t^f , and approaches
the value of 2 as a limit. For such pulses, for values of t^f greater than 1.0,
the quantity k/fQ can be approximated by the expression
f . 1 ♦ api (..»)
*0 *o
where f(O.Jl) is the value of Jftt) *t a time equal to one half the natural period
of the system. When expressed as a fraction of the total pulse duration, this
time Is equal to 0.5/(tdf). In the following table, the approximate and exact
values of k/fQ are compared for an initially peaked triangular pulse and for a
coalne function of oue -quarter of a cycle. The exact values for the latter
pulse were obtained from a plot included in Ref. 5«
V
Pulse
Cosine Pulse
Approx.
Exact
Approx.
Exact
1.0
1.50
1.57
1.71
1.82
1.5
1.67
1.72
I.87
1.91
2.0
1.75
1.79
1.92
—
2.7.3 Design Rules. For design purposes, the response spectra
for the absolute Mad— deformation of undaaped systems subjected to half-
cycle acceleration pulses without any discontinuities can be approximated as
shown in Fig. 2.11. For specific numerical applications, it is convenient to
plot this diagram on a four-way logarithmic grid similar to that given in
Fig. 2 .12, in which the scales are expressed in absolute units Instead of the
dimensionless ratios used up to this point. To a first approximation, the
spectrum nay be defined by the straight line segments ab, be, de, and the
curved segment cd. Improved accuracy can be obtained by use of the smooth
transition curve, as shown by the dotted line.
The spectrum is defined as follows:
(a) Along the horizontal line ab, the relative pseudo-velocity ▼
is equal to the mart mum value of the ground velocity.
(b) Along the diagonal line be, the acceleration A is approximately
equal to 1.5 times the mart mob ground acceleration.
(c) Along the diagonal line de, the acceleration A is equal to the
ground acceleration.
2-2 k
(d) The curve cd Is tangent to the line be and Intersects line da
at an angle, as shewn In the figure. The frequency corresponding to point d
is determined from the expression € f - 1.25, and that of point c can best
be estimated from the data given in Table 1. For a symetrical pulse, the loca¬
tion of c may be determined from Eq. 2.52.
For a discontinuous input function, the diagram must be modified
in accordance with the general observations made previously.
2.8 Synthesis of Spectra for a Sequence of Half-Cycle Acceleration Pulses.
In addition to providing a great deal of insight into the behavior of
the system, the detailed spectra of the type presented in Figs. 2.4 and 2.5 can
also be used to synthesize the response spectrum corresponding to a sequence of
half -cycle pulses. This possibility is described with reference to the full-
cycle acceleration pulse shown in Fig. 2.15, the individual pulses of which may
be of any shape for which detailed spectra are available.
The basic idea is to consider the motion produced by each half -cycle
pulse acting independently, and to coridlne the resulting maxirnim effects,
taking into consideration both the shape and duration of the individual pulses
and also the times at which these effects take place. Fig. 2.15 shows the
motion produced by each component pulse acting alone, along with the notation
used. The symbol denotes the maximum value of the acceleration produced
o,x
by the first pulse during forced vibration, l.e. in the interval t < t^, and
2 , denotes the corresponding residual maximal. The remaining symbols are
r,i
self-explanatory .
The absolute maximum response due to the actual pulse will naturally
occur in one of the following regions:
Region 1, corresponding to t < t^
Region 2, corresponding to t^ < t < t^
Region 5» corresponding to t > t^
2-25
Let Rj denote the magnitude of the maximum response for the Jth region, and
R be the absolute maximum response.
For the first region, R. Is evidently equal to X , , the latter value
being determined from the appropriate spectrum for the first half -cycle pulse
and the specified value of the frequency parameter t^f.
For the third region, R^ is obtained by a combination of the ampli¬
tudes of the tvo residual oscillations, x 1 and 'i 0, these quantities being again
r,x r yd
determined from the appropriate spectra with the appropriate values of t^f and
tgf . These asplltudes may be combined In a number of different vays, of vhlch
the following tvo appear to be the more appropriate:
(a) Take the numerical sum of 'i . and '£ _
r,x r,z
(2.3*)
(b) Use the expression
®3 = * ^Sr,2^
(2.35)
The first approach, vhlch assunes the tvo residual oscillations to be In phase,
obviously leads to an upper bound. The sign of the error committed by the
second approach, vhlch amounts to assuming the tvo residual oscillations to
be 90° out of phase, cannot be determined In general.
For the Intermediate region, the response Is computed by combining
the quantity 'i. 0 vlth the amplitude of the residual oscillation due to the
first pulse, x . . Tvo alternative procedures are noted vhlch are analogous to
those used for region 3.
An upper bound may be obtained by linear sqpexpoaltlon *a follows:
R a < * *rA (2-36)
vhere the signs are selected so as to yield the marlmim possible mmerleal value.
2-86
Alternatively, one may use the expression
*2= *2 + j <Xo,2 ‘ «2>2 + 'Sr,l)2 ' (2’37)
where (xq 2 - a^) approximates the oscillatory component of the motion induced
by the second pulse, and the square root quantity approximates the amplitude
of the oscillatory component due to both pulses.
As an illustration, consider an acceleration function composed of
a sequence of two half -sine waves such that t^/t^ ■ l/k and » t^ t\ - 1/3 ‘
Assume further that t^f =0.5; whence it follows that tgf ■ 1.5.
By entering Fig. 2.4 with the appropriate values of the frequency
parameter, one finds that
*0,1 ■ 1-5T ?o
S ,
r,l
*o,2 ■ 1,5 *2 “ °‘5 yo
ee
xr,2
± 1.57 yG
0
It follows that
’ '*0,1 ■ 1'57 yo
*2<
(-1.57 - 0.5) y0 . - 2.07 yo, *4- 2.36
[- 0.33 - / (0.5 - 0.33)2 + (1.57)2] V0 - - 1-91 Y0, ty 84. 2.37
and
*3
^(1.57 + 0) f0 = 1.57 y0, *7 Bq. 2.5*
V y<i.57)2 + 0 y0 - 1.57 y0, *t Bq. 2.35
The absolute marlrai value of the response is, therefore, R ■ -2.07
by linear superposition, and R ■ -I.91 by the square root rule. The latter value
happens to coincide with the exact value.
2-67
In Fig. 2.l4a the response spectrum for the acceleration function
considered in the preceding exeuqple is compared with the results obtained by
the two versions of the approximate procedure presented. A similar cosparl-
. son is made in Fig. 2.14b for a full-cycle sinusoidal function. As might have
been expected, the agreement is better in the first case where one of the
pulses dominates the response.
It must be noted here that the response of Idle system in the low-
frequency range can reliably be predicted by simple relations to be presented
later, and therefore this procedure need not be used for this frequency range.
The procedure is recommended especially for the computation of the absolute
i— -Hum value of A, and will be used for this purpose later.
2.9 Deformation Spectra for Undamped Systems Subjected to Half -Cycle Velocity
frulsoa
The pulses considered in this section are of the type shown in Fig. 2.2a
for which the areas under the positive and negative parts of the acceleration
function are equal. The system is presumed to have no damping and to be ini¬
tially at rest.
2.9.1 Low Frequency Systems . If the duration of the velocity pulse,
t^, is short in comparison to the natural period of the system, the maximum
ground displacement, yQ, will be attained before the mass of the system has had
an opportunity to respond, and the ground motion will literally be "absorbed"
by the spring. It is physically apparent that the first extremum value of the
deformation will occur approximately at t • t^, and will be nearly equal to the
negative value of yQ. The subsequent motion of the mass will be essentially
that of a fixed-base system subjected to an initial deformation -yQ. It
follows, therefore, that
2-28
o
u_
Bin
\in
- r.
Vx“yo
Sax"27
2 y.
and that the first values of and will occur approximately at
t - td 0.5T.
The limiting value of U for this case can also he determined from
Eq. 2.28b by ashing use of the analogy expressed by the second and third terns
in Eq. 2.17. Voting that A » p 2U, one obtains
▼ <P /«t) dt - p yfi
(2.38*)
or
£<1
yo~
nt yQ is expressed in te:
If the maxi bob ground displac
acceleration, Eq. 2.38 can be written alternatively as
of the ground
TJ < f‘M t dt (2.38b)
The latter Integral represents the none at of the acceleration diagram about
the end of the pulse.
2.9*2 Presentation and Discussion of Results.
a. Character! stl c s of Representative Spectra. In Pig. 2.15 are
given response spectra for the relative displacement U, the relative pseudo*
velocity ▼, and the pseudo-acceleration A, of a system subjected to a versed-
sine pulse of ground velocity. A sketch of this pulse and of the associated
acceleration and dlsplaceaent functions are Included in the figure. It must be
eaphaalzed that these curves are Interrelated by Eq. 2.21, and that if one of
then is known, the renalnlng two can be determined.
It can be seen that the def ornation U never exceeds the —
ground displacement, and that, for small values of t^f, it Is essentially’
2*9
equal to jQ, As far as the acceleration A Is concerned, at Large values of
tdf , It Is approximately equal to the maximum ground acceleration Y0> but the
peak value of A Is greater than VQ and occurs la the Intermediate range of
t^f values. These limiting values of U and A are In agreement vith those
discussed In the preceding section.
Of special significance is the relative order of magnitude of u/y0
and A/y0 for the extreme values of t^f . At small values of t^f , the amplifi¬
cation factors for A are a fraction of thoae for U, ehereas at the large
values, the order of the curves If reversed, and the amplification factors,
for U are a fraction of those for A. It would appear that, for low-frequency
systems, the maxi mini deformation is Insensitive to the details of the accelera¬
tion and velocity records, whereas for high-frequency systems, it Is Insensitive
to the characteristics of the lrput displacement. In the Intermediate range,
the response appears to he sensitive to the characteristics of both the velocity
and the acceleration traces. Because of the general shape of these curves, the
narlmna deformation of ms diem-frequency systems can more conveniently be
expressed In terns of T Instead of directly in terms of U. Similarly, for
high-frequency systems, ths quantity A Is a more convenient measure of the
maxi mam spring deformation than either U or T. It Is essentially for this
reason that T end A are used as alternative measures of U.
The maxi mm positive end the msxlmnn negative values of the spring
deformation are shown separately In Tigs. 2.16 and 2.17a In the fans of
acceleration spectra and pseudo- velocity spectra, respectively.
The following eharacterlstica of the curve* are worth acting.
(a) For values of t^f < 1, the mart mas defenatlom occur* during
free vibration, with the result that the positive and negative values of the
response are aannrleally equal.
2-^0
(t>) The absolute sail— value of the pseudo-velocity, VQ, occurs
during free vibration.
(c) For values of t^f >1, the maximum deformation during forced vibra¬
tion is equal to, or constitutes a good approximation to, the absolute mart man
value. Furthermore, the positive and negative values of the response are
close to one another, but this agreement is believed to be valid only for
symmetrical velocity pulses for which the positive and negative parts of the
ltqput acceleration have the same general shape and magnitude. If the peak
magnitudes of the two parts of the acceleration function are different,
will converge to the corresponding negative value. This condition is illus¬
trated in Fig. 2.17b which refers to a skewed versed-sine velocity pulse with
a rise time of 0.2^ t^. The ratio Of the minimum and maximum values of the
input acceleration function being l/3, the limiting value of It^J ■ fQ/5'
(d) For the pulses considered, the peak values of the acceleration
A for the forced vibration and the free vibration eras of the notion are dose
to each other. Pertinent data are summarised below for a class of skewed
versed-sine velocity pulses having rise-duration ratios of 1/2, l/k and 1/8.
Vv
td
Absolute Maximus
i Talus of A/Vc
1/2
3.25
3.21
lA
1.97
1.7*1
1/8
1.85
1.71
b. tffecte of Rise Tima and Discontinuities la Acceleration. The
spectra la Fig. 2.18 are for a f sadly of skewed versed-sine velocity pulses
with rise times hanging from 1/2 to l/8 the duration of the ydse.
2-31
It can clearly be seen fron this figure that for values of t^f less
than one the effect of rise time is almost Imperceptible . Similarly, at the
right hand end of the diagram, the limiting value of A for each curve can be
shown to be equal to the maximum input acceleration fQ. By virtue of the
fact that the magnitude of the ground acceleration for a fixed value of the
— Tina ground velocity Increases with decreasing rise time, on a plot such
as that given in Tig. 2.18, the location of the limiting value of A shifts
to the right as tr ^/tg decreases. Thus, the principal effect of a decrease
in the rise time is to Increase the width of the nearly flat portion of the
V-spectrum.
In Pigs. 2.19 through 2.22c are given deformation spectra for several
velocity pulses the derivatives of which are discontinuous functions. The
velocity pulses considered Include a symmetrical parabolic pulse (Pigs. 2.19
and 2.20), a skewed sinusoidal pulse with a rise tine equal to one third the
total duration (Pig. 2.21), and a series of triangular pulses with different
rise-duration ratios (Pigs. 2.22a through 2.22c). The velocity pulses and
the corresponding acceleration histories are shown in the inset diagrams. The
values of t^f below which the absolute maxi mas response consistently occurs
during free vibration are also indicated.
For values of t^f between zero and a value slightly greater than that
for which Y is maximum, the spectra presented in these figures are almost
identical to those presented earlier, verifying the prediction that, for flexible
systems, the deformation is dependent on the shape of the ground dis¬
placement alone, rather than on the shapes of the corresponding velocity or
acceleration traces. In each case, the spectrun is bounded on the left by a
line of constant displacement equal in magnitude to the maximum ground displace¬
ment. For values of t.f less than that for which Y/i ■ 1, the maximum error
U O
due to taking U ■ yc is for all practical purposes negligible.
2-32
I* contract, for values of t^f >1, the mgnitude and general appear¬
ance of the curves are influenced to a rather significant degree by the detailed
features of the input notion. In Figs. 2.20 and 2.21 the Uniting value of A
is equal to twice the aarlnti ground acceleration, and in Figs. 2.22 it is
equal to twice the narlnun value of the discontinuity in the input acceleration
function. These Uniting values are la agreenent with those predicted by the
procedure described in Art. 2.7.2b. In Fig. 2.22c the curve for tr ^/t^ ■ 0
approaches asymptotically the line V/yQ - 1, because the maximal input accelera¬
tion is infinite in this case, and the velocity function approaches a step pulse
of infinite duration.
For an input acceleration without any discontinuities, the response
of a high-frequency system nay be considered to be the sane as that obtained
under static conditions if Eq. 2.30 is satisfied for each component pulse in
the input acceleration. However, if the aaplltudes of the Individual pulses
are significantly different from one another, it nay be sufficient to satisfy
this relation only for the pulse with the greatest ordinate, since the effect
of the remaining pulse or pulses nay be negligible.
c. Mariana Values cf V and A. The narlnun values of V, for the
velocity pulses considered in the preceding sections and for several additional
pulses considered in Ref. 2, are listed in Table 2 together with their corres¬
ponding values of t^f . The results for the pulses Identified with an asterisk
correspond to narlna that occur during free vibration, but these narlna are
expected to be equal to or very close to the absolute naxlnun values. Pulses
3a and 5b are defined by Xq. 4.6 of Ref. 2 as the product of a versed sloe
function, a skewing constant, and a decaying exponential function.
It can be seen that the value of ranges between 1 and 2, the
lower bound corresponding to n rectangular velocity pulse of infinite duration,
2-53
and the upper hound to a rectangular pulse of finite duration. These limiting
values suggest that, for velocity pulses of other shape, the smaller values of
V would occur for pulses having a sharp rise and a gradual decay, and
that the larger values, would correspond to pulses having a sharp rise, a sharp
decay, and a fairly flat Intermediate region. It follows further that, for
symmetrical pulses, the values of V0/t0 can he expected to he greater than
those for unsymnetrlcal pulses, and that among syanetrlcal pulses of the same
duration and the same peak value, the greater values of V q/yo would correspond
to the pulse having the shortest effective rise time combined with the flattest
top. These conclusions are substantiated hy the numerical data presented in
Table 2. The effective rise time, € , defined In a manner analogous to that
* r
used for a half -cycle acceleration pulse, is given hy the equation
l
rfr
(2.3
In the absence of specific Information about the shape of the velocity pulse,
the value of VQ for the unsymmetrlcal velocity pulses encountered in ground
shock problems may he taken as 1.3 times the maximum Input velocity.
It may he recalled that In discussing the effects of half -cycle
acceleration pulses, it was noted that the maxi mum value of A depends primarily
on the effective rise time of the pulse, and that the detailed shape of the
pulse. Including Its decay time, were relatively unlaportant. That the signi¬
ficant parameters for an acceleration Input are different from those for a
velocity Input can best he appreciated hy considering the response of high-
frequency systems to a rectangular forcing function. For an acceleration
input, the amplification factor for Is one. Irrespective of the duration
of the pulse, whereas for a velocity Input, the a^pllflcator f motor for ?Q
Is one only for a pulse of infinite duration, and becomes two for a pulse of
finite duration.
2-3*
In Table 2, the values of t^f corresponding to VQ range from 0.50
to 1.9, with the majority of the values being of the order of 0.7. The maxi¬
mum value is obtained for the decaying skewed versed-sine pulse, No. ta, for
which it is physically apparent that the "effective duration", t , which
o,v
excludes the low-intensity tail end of the pulse, is shorter than the actual
duration. The location of VQ can more reliably be expressed in terms of the
effective duration parameter t f , the values of which, as can be seen from
o,v
the table, are considerably less dependent on the details of the pulse shape
than are those of the parameter t^f . The quantity tQ y, defined in a manner
analogous to that presented earlier for an acceleration pulse, is given by
the expression
t
■Jr*.
(2 .kOtt)
In the absence of detailed Information about the shape of the velocity pulse.
Vq may be considered to occur at a value of
t f s 0.8 or t.f s 0A
O.T d V
' 'av
(2Aob)
For the pulses considered, the values of Aq /yQ range from a inaxlnun value
of 4 to a value of less than 2. In general, the larger values are obtained for
the acceleration pulses for which the positive and negative half -cycles are of
the same shape and duration (l.e., for symmetrical velocity pulses). In the
following table, the exact values of for the class of skewed versed-sine
velocity pulses considered are compared with the values obtained by application
of the two verslona of the approximate procedure described in Section 2.8. For
the values given in the third column, the contributions of the individual pulses
were combined linearly, and for the values given in the fourth column the square
root rule was used. The agreement between the exact and the approximate values
is considered to be quite adequate for all practical applications.
2-55
Maximum Values of A /y
o' *0
Exact
Approximate
1/8
I.85
I.85 1.84
iA
1*97
2.17 2.06
1/2
5.50
5.46 2.86
Strictly speaking, the value of AQ depends not only on tharelative
amplitudes end durations of the individual pulse* in the Input acceleration
function, hut also on the pulse shapes;. themselves, as nay he appreciated
fron the discussion presented in Section 2.7*2. However, in the absence of
information shout the detailed shape of these pulses, the value of Aq nay he
taken approximately as 1.5 times the numerical sun of the maximal and minimal
values of the input acceleration.
The location of A can hast he defined in terms of the duration t. „
o x,n
of the arMiwiuvfc acceleration half -cycle rather than the total duration of the
pulse. The quantity t. is, of course, equal to shorter rise time in the
x,a
associated velocity pulse, tr y. For continuous functions, Aq may he considered
to occur at a value of
t. f = 0.6 (2.4l)
2.9.5 Design Rules. For design purposes, the deformation spectra for
systems subjected to half -cycle velocity pulses may he approximated by the diagram
given in Fig. 2.25, provided the ground acceleration 1* a continuous function.
To a first approximation this spectrum may he defined by the straight line seg¬
ments ab, he, cd, ef and the curved segment de, as follows:
(a) Along the diagonal line ab, the displacement U Is equal to the
— ■rH—i value of the ground displacement.
2-56
(b) Along the horizontal line be, the re la tire pseudo- velocity ▼ is
equal to 1.5 tines the narlnum ground velocity. If the detailed shape of the
Input velocity is known, a more precise estlnate for this upper bound on V nay
be obtained from the data presented In Table 2.
(c) Along the diagonal line cd, the acceleration A Is equal to 1.5
tines the sun of the absolute values of the narlnun and alnlnua ground accelera¬
tions. if the input function Is known exactly and response spectra for the
component pulses are available, a somefchat better estlnate of this value of A
nay be obtained by the procedure described in Section 2.8.
(d) Along the line ef , the acceleration A Is equal to the
ground acceleration.
(e) The curve de is tangent to the line cd and intersects the line
ef at an angle, as shown In the diagram. The frequencies corresponding to
points d and e are determined approximately frost the expression shown In the
figure. The frequency for point d should not be snaller than the frequency
corresponding to point c of the diagram.
(f) Tbs transition curves shown In dotted lines are tangent to the
straight line secpents at points g, h and d. Point g corresponds to a value
of V - yQ, and point h corresponds to a frequency determined from Sq. S.kOb.
The latter frequency should not be greater than that corresponding to point c.
2.10 Deformation Spectra for Undamped Systems Subjected to *M^-Cycle
BTsplacenent Pulses and Pulses with Partial Recovery
The pulses considered In this section are of the type shown In
Pigs. 2.2c and 2.2b. AsVbefore, the system Is considered to have no damping
and to be Initially at rest.
2.10.1 Low Frequency Systems. Prom a physical argonaut entirely
analogous to that used In Section 2.9*1* one concludes that for low-frequsney
2-37
systems the first extremum value of the deformation will occur at or near
the instant that the ground attains its maximum value and will be approx! -
itely equal to the negative value of yQ, i.e.
umln " ‘ yo - - d
at t = t .
(2.42)
The second extremum will occur after termination of the pulse . By virtue of
the similarity of Eqs. 2.3 and 2.5, the absolute displacement of the system
for t > td can be determined from Eq. 2.27b by replacing x8t(t) by y(t), as
follows
*(t) = P £ J'jKt) drj sin pt - ptd yav sin pt
(2.43a)
where y is the average value of the displacement in the interval between
® v
0 and t&. Equation 2.43a also represents the relative displacement u(t),
since y(t) - 0 for t > t^. It follows then that
or
"he first values of u^T and occur at a time roughly equal to one-half
the natural period of the system. Incidentally, Eq. 2.44 could also have been
obtained from Eq. 2.38a by utilising the analogy expressed by the first two
terms in Eq. 2.17-
For a ground displacement y(t) with partial recovery, the relative
displacement for t > td is given by the expression
u(t) s pt4 yav sin pt - yf cos(p(t-td)l,
in which the first term represents the contribution of the pulse within
°5*<td, and the second tern represents the contribution of the residual
or final displacement of the ground, yf . For small values of ptd, taking
sin ptd ■ ptd and coa ptd - 1, one obtains
2-38
whence
u(t) » ptd[yav - yf ] sin pt - yf cos pt,
(2.45)
For a displacement pulse with complete recovery, the absolute maxi¬
mum value of the deformation, U, is the numerically larger of the values given
by Eqs. 2.42 and 2.44, and for a pulse with partial recovery, it is the larger
of the values given by Eqs. 2.42 and 2.45. For a displacement pulse with com¬
plete recovery, Eq. 2.41 governs for values of
t f < JL 1£L
4 2* *.v
2.10.2 Presentation and Discussion of Data. In Fig. 2.24 are
presented response spectra for the and minimum deformations of undamped
systems subjected to a half-cycle displacement pulse. The acceleration diagram
of the input motion consists of a sequence of three half-sine pulses of the
same amplitude and of durations t^, 2t^ and t^, respectively, as shown in the
inset diagram. The deformation, uwmiv, corresponds to an extension of
the spring and is a positive quantity, whereas uM<n corresponds to compression
and is a negative quantity. It should be noted that the abscissa in this figure
is the quantity 2t.jf Instead of the quantity t^f used in previous figures.
The quantity 2t^ is also equal to the duration of each velocity pulse and to
the rise time of the associated displacement function. Included in this figure
as dotted line curves are also the results obtained from Eq. 2.42 and from the
right member of Eq. 2.44.
In Figs. 2.25a and 2.25b are presented similar curves for a versed-
sine displacement pulse and for a skewed versed-sine displacement pulse vith a
rise-duration ratio of t^/t^ ■ l/4. The upper envelopes of these curves are
coopered in Fig. 2.26 with the corresponding curve for a displacement pulse of
2-59
the sane family but with a value of t^/tj - l/8. The displacement pulses
considered In these figures are identical to the velocity pulses considered in
Figs. 2.17a through 2.18.
The following characteristics of the curves are worthy of note:
(a) Unlike the spectra for the half- cycle velocity pulses considered
previously* which were bounded by a value of U equal to the maximum ground
displacement* the spectra presented in this section have values of U exceeding
the maximum ground displacement over a considerable range of the frequency
parameter.
(b) For low-frequency systems* the results obtained from Eqs. 2,42
and 2.44 are in good agreement with the exact values. Hote* in particular*
that the approximate results define with reasonable accuracy the initial position
and the Initial slope of the "hump" on the left-hand portion of the diagram. In
the following table the values of t^f corresponding to this break in the spectra
are coapared with the values obtained from Eq. 2.46 for the family of skewed
versed -sine displacement pulses.
*A
Value of t^f
Exact
From Eq. 2.46
1/2
0.15
0.16
1/4
0.084
0.080
1/8
0.042
0.059
(c) For the motion considered In Fig. 2.24 the limiting value of
A/y- for high-frequency systems Is 1.0* whereas for the motions considered in
Figs. 2.2? it Is equal to 2.0. This is due to the fact that the acceleration
function: of the first motion is continuous* whereas of the second motion It Is
discontinuous. These limiting values are In agreement with those predicted by
the procedure described in 8ection 2.7.2b.
(d) The absolute maximum value of the deformation, Uq, occurs during
free vibration, and the maximum value of the pseudo-velocity, VQ, either occurs
during free vibration (as in Figs. 2.24 and 2.25a), or is a close approximation
to the corresponding maximum obtained during free vibration (as in Fig. 2.25b).
It may be recalled that for the half-cycle velocity pulses considered in
Section 2.9 similar results vere obtained for the quantities VQ and Aq.
Accordingly, the analogies given in Eq. 2.17 are applicable, and it follows
that the value of UQ/yo for a half-cycle displacement function most be equal to
the value of VQ/yo for a velocity input of the same shape, and they must occur
at the same values of t.f. Similarly, the coordinates of VQ/yo for the
displacement input may be considered to be the same as those for ^Q/VQ for the
corresponding velocity input. That this Is indeed true can be verified by
comparing the corresponding coordinates of the spectra presented In Figs. 2.17
and 2.25. The magnitude of Aq for the displacement pulses cannot be obtained
by analogy of the results presented previously, but it Is clear from the
material already presented that this quantity depends primarily on the number
of half-cycles in the Input acceleration, on the degree of regularity of the
Individual pulses, and, to a lesser extent, on the shape of the individual
pulses. If the spectra corresponding to the component pulses are available,
then the value of Aq may be determined with good accuracy by use of the procedure
described in Section 2.8. In the absence of detailed Information about the
characteristics of the ground acceleration, the approximate design rule given
in the next section nay be used.
In Fig. 2.27 are given deformation spectra for a half-sine displacement
pulse. It is important to note that, whereas the left-hand portions of these
spectra are quite similar to those presented In Figs. 2.24 and 2.25a, a result
that might have been anticipated from the similarity of the three displacement
2-41
functions , the right-hand portions differ radically. It should be apparent
that the medium-frequency and high-frequency regions of a deformation spectrum
are functions of the detailed or "microscopic" features of the displacement
function, which are generally difficult to ascertain from the displacement
diagram Itself. These features are most clearly depicted in the velocity and
acceleration diagrams of the input motion. In Fig. 2.27 the spectrum approaches
a horizontal asymptote because the velocity diagram of the input motion is a
discontinuous function (i.e. the acceleration function has Infinite discontinuities).
By utilizing the procedure described in Section 2.7.2b and the analogy
expressed by the second and third terms in Eq. 2.11, it can readily be shown
that as t.f « the maximum and minimum values of the velocity of the mass, x,
approach a value equal to twice the maximum ground velocity. This relation is
also valid during free vibration. But, since y(t) - 0 for t > tfl,
W - 1*1 * M
whence it follows that
l*Wl " l^miJ “ 2 K
Seme data for displacement pulses with partial recovery are given
in Ref. 8.
2.10.3 Design Rules. For design purposes, the deformation spectrum
corresponding to a half-cycle displacement pulse, or a pulse with partial
recovery, may be approximated by the diagram abcdefgh, as shown in Fig. 2.26.
For iaqproved accuracy, the portion of the diagram between points b and f may
be replaced by smooth transition curves as shown by the dotted lines. The
time histories of the displacement, velocity and acceleration of the ground
2-1*2
are considered to be continuous functions. The characteristics of this diagram
are as follows:
(a) Along the limiting lines ab and gh, the relationship between
the input and response quantities are the same as for the corresponding lines
ab and ef in Fig. 2.23* Furthermore, the frequency corresponding to point g
is the same as that for point e in Fig. 2.23.
(b) For a displacement pulse with complete recovery, the displacement
U along line be is given by the right-hand member of Eq. 2.44, and for a
displacement pulse with partial recovery, it is given by Eq. 2.45.
(c) Along line cd, the displacement U may be approximated by the
equation
U * yQ [l.5 - 0.5 (2.47)
(d) Along de, the relative pseudo-velocity V 1b equal to 1.5 times
the sum of the absolute values of the maximum and minimum ground velocities.
This relationship is the same as that between the response acceleration A and
the input acceleration for line cd in Fig. 2.23* If the velocity of the
ground is known accurately and the deformation spectra corresponding to the
component pulses of the velocity function are available, then an improved
estimate of this max Inca value of V may be obtained in a manner analogous to
that described in Section 2.8.
(e) Along line ef, the acceleration A is proportional to the
maviwniw input acceleration, the ratio of proportionality depending on the
degree of regularity of the input function. If the durations for the indivi¬
dual pulses of the ground acceleration are approximately equal to each other,
then the value of A along this line nay be determined from the expression
2-43
(2.1*8)
Ao 5 W t l(yo)jl
>1
where (y ) Is the amplitude of the Jth half -cycle, and n is the number of
O J
half-cycles present. If the amplitudes of the individual pulses are of the
same order of magnitude but their durations differ appreciably, then the result
obtained from this equation may be quite conservative. If the deformation
spectra for the component pulses are available, an improved estimate of Aq may
be obtained by the procedure described in Section 2.8.
(f) The frequency corresponding to point f is determined from
Eq. 2.4l, where t. should be interpreted as the average duration of the
x,a
acceleration half- cycles contributing over fifty percent of the value of AQ.
The curve fg is similar to the curve de in Fig. 2.23.
(g) Hie transition curves represented by the dotted line are tangent
to the corresponding straight line segments at points b, 1, J and f . Frequently,
the lengths of the straight line segments between b and f are small, and the
transition curves can be drawn vlthout having to evaluate the location of
points 1 and J. When this is not the case, the frequency corresponding to
point 1 may be determined from the following expression, obtained by analogy to
Eq. 2.40b,
t f = 0.4 (2.48x)
°av
and the frequency corresponding to point J, may be determined from the expression
t^vf - 0.6 (2.49)
where t. is the duration of the velocity half-cycle with the maximum amplitude .
i;"
Strictly speaking, Eq. 2.40b Is applicable only to displacement pulses with
2-44
complete recovery, but, for want of any better information. It may also be
used for pulses with partial recovery. It should be noted that points 1 and
t, which are analogous to points h and d In Pig. 2.23, should be within the
limits of the lines cd and de, as shown in the figure.
2.11 Deformation Spectra for Undamped Systems Subjected to Full -Cycle
Displacement Pulses
For the typical full-cycle displacement pulse shown in Fig. 2.29,
let (y ), and (y )„ denote the numerical values of the first and second
extremums, and (t )^ and (tQ)2 denote the corresponding times. In addition,
let t^ denote the total duration of the function, and t^ and denote the
durations of the first and the second half -cycles .
2.11.1 Low Frequency Systems. As previously explained, in the
Interval 0 < t < t^ the time history of the deformation for systems with small
values of t^f may be taken equal and opposite to that of the ground. Accord¬
ingly, the first extremum value of the deformation, (u^, will occur at
t • (tQ)^, and will be given by the expression
<tto>l 2 ' (2*5°)
The second extremum, (uq)2, will occur at t * (t0)2 will be given by
the expression
(uc)2 s (y0}2 <2.51»)
Eq. (2.31a) is valid for very soft systems, or more precisely, for
values of t^f — •* 0. For somewhat stiff er systems, a better estimate of (uq)2
may be obtained from the expression
(u0)2 8 (lKr)1 •!«» Ip(*0)2] ♦ (f0)2 (2.51b)
where, for values of p(t0)g greater than x/2, the term sln[p(t0)2l should be
taken as one. The quantity (7aT)^ 1* this equation denotes the average value
2 -*$
of the displacement between t * 0 and t ■ t^, and the term Involving (y
represents the simple impulse theory approximation to the contribution of the
first pulse.
The third extremum value of the deformation, (uQ)y occurs during
free vibration. For a displacement function for which the areas under the
positive and negative parts are equal, the value of this extremum is given by
the following expression which, by virtue of the similarity of Eqs. 2.3 and
2.4, may be obtained directly from Eq. 2.38b by replacing 'yi t) by p2y( f)
*d
KU0)3I < p2/ y(T)TdT (2.52)
0
The absolute maximum value of the deformation, U, is the numerically
greatest of the values given by Eqs. 2.50, 2.51b and 2.52. When (y©)2 > (y©)^
the second extremum is numerically greater than the first, and Eq. 2.50 need
not be considered if only the absolute maxi imam value of the deformation is
required.
2.11.2 Presentation and Discussion of Data. In Fig. 2.30 is given
the deformation spectrum corresponding to a full-cycle displacement function
which is Identical in shape to the velocity diagram of the half-cycle displacement
pulse considered in Fig. 2.24. In the extreme left-hand portion of the figure,
the first three extremum values of the deformation are shown separately, along
with the corresponding results obtained from the approximate equations of the
preceding section. Similar information is presented in Figs. 2.31a through
2.32 for a family of displacement functions composed of a sequence of two half-
slne waves. These functions are identical to the velocity diagrams considered
in Figs. 2.25a through 2.26.
2-46
The following results are worth noting:
(a) For low-frequency systems, the results obtained by the
approximate equations are in reasonable agreement with the exact values.
(b) The magnitude and location of the absolute may-imum deformation
Uq for the spectrum presented in Fig. 2. 50 are identical to those of Vq for
the spectrum presented in Fig. 2.24. This result is a consequence of the
analogy expressed by the first two terms in Eq. 2.17. The same is also true of
the coordinates of UQ in Figs. 2.51a through 5*52 and the coordinates of Vq
of the corresponding spectra in Figs. 2.25a and 2.26.
(c) The coordinates of the absolute maximum pseudo-velocity, V ,
for the spectra in Figs. 2.50 through 2.52 are identical to, or approximately
equal to, the coordinates of the absolute maximum pseudo-acceleration, Aq, of
the corresponding spectra in Figs. 2.24 through 2.26. The approximation in
this case arises from the fact that these maxima generally do not occur during
free vibration, and the analogy expressed by the last two terms in Eq. 2.17 is
not strictly applicable.
(d) In contrast to the spectrum in Fig. 2.50 which at high frequencies
approaches a diagonal asymptote, the spectra in Figs. 2.51 and 2.52 approach
a horizontal asymptote. This difference can best be explained by reference to
the acceleration diagrams of the ground motion. In Fig. 2.50 the acceleration
function has finite discontinuities, and the limiting value of the deformation
spectrum is A/yQ • 5.0, as might have been predicted by the procedure presented
in Section 2.7.2b. On the other hand, the curves in Figs. 2.51 and 2.52
approach a horizontal asymptote because the velocity diagram of the input motion
is discontinuous, l.e. the acceleration function has Infinite discontinuities.
If both the ground velocity and the ground acceleration were continuous, the
limiting value of A would have been equal to the maximal ground acceleration.
2.11.3 Design Rules. On the basis of the Information presented, it
is concluded that the response spectrum for a full-cycle displacement function
may be approximated by the diagram shown In Fig. 2.33. Both the displacement
function and its first two derivatives are assumed to be continuous .
Along the straight line aa*, the displacement U is equal to the first
wtfnHwnm ground displacement. Curve a'b is defined by Eq. 2.31b, and point b
Is the intersection of this curve and the curve represented by the right hand
member of Eq. 2.32. Along the straight line cd, the displacement U Is equal
to 1.3 times the sum of the absolute values of the maximum ground displacements.
Point 1, which is equivalent to point J in Fig. 2.28, corresponds to a
frequency determined from the expression
tx,df S 0.6 (2.53)
where t, . Is the duration of the displacement pulse with the maxi mum enroll -
l,d
tude. This point should lie to the left of point d. The velocity V along the
horizontal line de may be approximated by the expression
V - 1.5 £ (2#54)
>1
If the deformation spectra corresponding to the component velocity pulses are
available, a more accurate estimate of V may be obtained with the aid of the
procedure described in Section 2.8. The remaining features of this spectrum
are similar to those of the spectrum given In Fig. 2.28.
2.12 Relationship of Computed Results to Field Test Data
The approximate rules presented In the preceding section are substan¬
tiated by the results of the field tests analyzed In Ref. 1. Included In this
reference are response spectra for systems with a very small amount of damping
(0.3 percent critical) subjected to the horizontal and vertical components of
2-48
the ground motions measured In a number of field tests. These spectra are
given in dimensionless plots similar to those used in the present study. The
relative pseudo-velocity V is normalized vlth respect to the so-called -'velocity
Jump" which may be considered to be equal to the maximum ground velocity, and
the frequency scale is taken as the product of the natural frequency of the
system and the "duration" of the velocity pulse. The latter quantity is taken
as the time from arrival of the velocity Jump to the first zero value of the
ground velocity.
Three classes of spectra are distinguished in Ref. 1 depending on
the direction of the motion and the distance of the point under consideration
from ground zero. These correspond to:
1. Vertical motions in the superselsmlc region of the blast.
2. Vertical motions in the subselsmic region of the blast, and
3. Horizontal motions in general.
From the discussion in Section 2.6 it follows that the spectra in
Item 1 should be compared with those for half -cycle velocity pulses, and the
spectra of Item 2 should be compared with those for half -cycle displacement
pulses or for displacement pulses with partial recovery. Finally, the spectra
in Item 3 should be compared with those for half-cycle displacement pulses and
possibly for displacement pulses having both positive and negative parts. On
making these comparisons, one finds that the agreement between corresponding
spectra is in general very good.
The important features of the spectra given in Ref. 1 are as follows.
For vertical motions in the superselsmlc region of the blast, the values of
U are consistently smaller than the maximum ground displacement, and the maxi¬
mum value of V is about 1.3 times the maximum ground velocity. It is parti¬
cularly noteworthy that, in the regions of the spectrum where U pr V nay be
2-49
considered to be constant, the results fall vithln a very narrow band, whereas
for the high-frequency region of the diagram the "scatter" is considerable.
These trends correspond almost exactly to those of the spectra for half-cycle
velocity pulses presented in Section 2.9, as may readily be appreciated by
referring to Figs. 2.18 or 2.22c. The "scatter" in the high frequency range
emphasizes the difficulties Involved in specifying precisely the value of the
maximum ground acceleration. The value of t^f corresponding to the peak value
of the spectra in Ref. 1 ranges between a value of 1 and 3> from which it may
be inferred that the dominant velocity pulse of the ground is highly unsymaetrical.
For vertical motions in the subseismic regions of the blast and for
horizontal motions in general, the maximum values of U in Ref. 1 range from
1.5 to slightly more than 2 times the maximum input displacement, the greater
values corresponding to horizontal motions. The maximum values of V lie
between 2 and 2.5 times the maxi mum ground velocity for motions in the vertical
direction, and between 2 and 2.8 for horizontal motions.
2.13 Deformation Spectra for Damped Systems
In Figs. 2.34 through 2.43b are given deformation spectra for danqped
systems with coefficients of viscous damping between zero and 100 percent
critical for the following classes of ground motion:
(a) A family of skewed versed-sine velocity pulses with rise-
duration ratios cf l/2, l/4 and l/8 (Figs. 2.34 through 2.36)
(b) The following half -cycle displacement pulses: A pulse for
which the acceleration diagram consists of a sequence of three
half -sine waves of the same amplitude but different durations
(Fig. 2.37)> * family of displacement pulses having the shape
of the velocity pulses considered under item (a) (Figs. 2.38
through 2.40), and a half-sin* displacement puls* (Fig. 2.41).
2-50
(c) The following full-cycle displacement functions: A displacement
function having the shape of the velocity diagram for the first
motion considered under item (b) (Fig. 2.42), and two functions
composed of a sequence of two half-sine waves each (Figs. 2.42a
and 2.42b).
It can clearly be seen from these figures that the overall effect
of damping is to reduce the magnitude of the maximum deformations, and to
smooth out the humps and undulations of the spectra. It is also clear that
the extent of the reduction is generally different for the different frequency
regions, and that, within a given range of frequencies, it is different for
the different ground motions.
For the simple pulses considered, the effectiveness of damping in
reducing the magnitude of the maximum deformation can be related to:
(a) the number of oscillations that the system undergoes before
attaining its maximal deformation, and
(b) the amplitude of the oscillatory component of the response.
The latter component corresponds to the solution of the homogeneous part of
the governing differential equation of motion.
In general, other things being equal, the greater the number of
oscillations or the amplitude of the oscillatory component, the greater is the
reduction achieved with a given amount of damping.
In the low-frequency region of the spectrum, the effectiveness of
damping is generally small because the maximum value of the deformation is
reached at a small fraction of the natural period of the system. Since the
deformation in this region occurs at or near the instant that the ground
displacement attains its maximum value, it follows that in comparing the spectra
for the different ground motions considered, the comparisons should be made for
fixed values of where tf ^ is the rise time to the peak ground displacement.
2-51
fig. 2.36 shows that, for snail values of the frequency parameter,
damping has a greater effect on the positive deformations than on the
corresponding negative deformations. This condition arises from the fact that
u^ir| corresponds to the first extremw, whereas v^aax corresponds to the second.
Similar results are indicated in Pig. 2.40 for a versed -sine dlsplacesmnt pulse,
but in this case the reduction for is comparatively less pronounced than
in the preceding case. This difference can again be explained in terms of the
times at which the respective ■**“<■* occur. For the versed-sine velocity pulse
considered in Fig. 2.36, uw>r occurs approximately at one-half the natural
period of the system after the time of maximum ground displacement, whereas for
the versed- sine displacement pulse considered in Fig. 2.40, it occurs at one-
quatter the natural period of the system. These values were noted before in
connection with undamped systems.
In the high-frequency region of the spectrum, the effect of damping
depends on the amplitude of the oscillatory component of the motion, and this
amplitude depends, in turn, on whether the ground acceleration is a continuous
or a discontinuous function. For the input functions considered in Figs. 2.34
through 2.37, for which the ground acceleration is continuous, the effect of
damping can be seen to be negligible. In contrast, for the pulses considered
in Figs. 2.38 through 2.40, for which the ground acceleration is discontinuous,
the maximum reductions achieved are of the order of 30 percent for values of
0 > 1.00. The reductions for a given amount of damping are even greater for
the full-cycle displacement pulse considered In Fig. 2.42. For the pulses in
Figs. 2.38 through 2.40, the absolute mart mum value of the deformation corresponds
to the first STtraanim value, and It may be approximated by the following
expression, which Is applicable to a step acceleration function of long duration.
(2.55)
This equation Is valid for values of 0 less than one. lote that Its first
term, vhich corresponds to the particular solution of the governing differential
equation of notion, Is Independent of damping and equals the maximum input
acceleration. Equation 2.55 la also applicable to the full-cycle dlsplacsaent
pulse considered In Fig. 2.42, provided the aaount of (leaping In the system is
sufficiently large such that the first extras— value of deformation represents
the absolute maxis— deformation. The results presented indicate that this
condition holds true for values of 0 equal to or greater than about 0.05>
In the high frequency region of Fig. 2.40, It is of Interest to note
that, whereas for undamped and for critically damped systems, and u^1n
are numerically equal to each other, for systsms with 0 - 0.20, u^<n is
numerically greater than u^v. This result can be explained with reference
to Fig. 2.10e vhich shows the input acceleration together with the response
acceleration for a high frequency undamped system. It can be seen that the
first maximum positive acceleration (corresponding to occurs at a very
early stage of the motion, whereas the first maximal negative acceleration
(corresponding to u^,) occurs near the middle of the pulse, after the system
has executed several cycles of oscillation. Although for an undamped system
these two maxima are z— erlcally equal, for a system with damping u^in governs
because it occurs earlier than u^y. For a critically damped system, the two
maxima are nearly the same because the oscillatory component of the deformation
Is negligible In this case, and the remaining component, vhich is proportional
to the input acceleration, has the same positive and negative parts.
For the displacement pulses considered In Figs. 2.45a and 2.45b, the
maximum deformation of a high-frequency undamped system occurs mostly during
2-53
free vibration, following the second discontinuity in the velocity diagram.
Tor a system with a substantial amount of damping, however, the oscillatory
ccaponent of the motion induced by the first discontinuity is generally damped
out by the time the second discontinuity is applied, with the result that the
maximum deformation during free vibration is no larger than that attained
during forced vibration. Under these circumstances, the deformation of
the system can be approximated by the following expression that gives the effect
of a sudden velocity change without rebound. The equation la valid for values
of 0 < 1.
— 6 - t*,-1 £12] (2.56)
The results obtained from this equation are found to be in good agreement with
the exact results for systems with values of t^f greater than about 2 and
values of fi greater than about 0.10. For smaller values of 0, Eq. 2.56 defines
the lover envelope of the response spectra.
Excepting ground motions for which the acceleration and/or velocity
diagrams are discontinuous, it can be said that the effect of damping is
greatest in the medium-frequency region of the spectrum, both because the
amplitude of the oscillatory coaqponent of the response is appreciable in this
case and because the maxima deformation of the undamped system is usually
attained near the end of the disturbance, after the system has undergone one
or more cycles of oscillation. For this region, the greater the periodicity
of the Input motion, the greater is the reduction in the peak value of
deformation achieved with a given amount of damping. Of course, the response
of the elastic system inereames with Increasing periodicity of the input motion.
These conditions are Illustrated In Figs. 2.44a through 2.44d where
deformation spectra are presented for systems subjected to velocity functions
composed of from one to four parabolic pulses of equal amplitude and duration,
as shorn in the inset diagrams. Hote that, vhereas the peak values of the
undamped spectra increase in almost direct proportion to the number of velocity
pulses in the input motion, the corresponding values of the spectra for highly
damped systems remain virtually unchanged. Bote, in particular, that the
curves in Figs. 2.44b through 2.44d are almost identical to each other for
values of 0 > 0.5. In these cases, the order of the controlling maximum is
the same, and it corresponds to an early, usually the first or second, maximum.
It should finally be noted that the peak values of U, V and A for elastic
systems are in good agreement with the approximate rules that have been presented.
2.14 Deformation Spectra for a Combination of Simple Pulses
The information in this section is intended to illustrate the
manner in which the deformation spectra for simple pulses presented in the
preceding sections nay be modified by the effect of high frequency oscillations
that may be superlsposed on the main pulses. The procedure used to arrive at
this information is approximate, and the results are mainly of qualitative
significance.
Let V1 and Vg denote the values of the pseudo-velocity corresponding
to the primary and the secondary pulse of the ground motion, respectively, and
let V be the corresponding value for the combined pulse. The value of V nay
then be determined by a procedure analogous to that presented in Section 2.8,
or more simply by taking the sum of the may 1mm contributions of the component
pulses, l.e.
That the value of V determined from this equation may he considerably greater
than the actual value, may be appreciated by noting that Eq. 2.57 la Independent
of the relative position of the component pulses.
In Figs. 2.45a and 2.45b are given deformation Spectra determined by
application of Eq. 2.57 for a combination of two versed-sine velocity pulses,
as shown in the inset diagrams. The quantities a^ and ag in these figures
denote the maximum accelerations of the primary and the secondary pulse,
respectively, and v^ and denote the corresponding maximum velocities. The
total duration of the primary pulse is denoted by V and that of the secondary
pulse by tg. In these figures, both the frequency parameter used as abscissa
and the response quantities are normalised with respect to the relevant dimensions
of the primary pulse.
Figure 2.44a illustrates the method of computation for a combination
of versed-slnce pulses with tg/t^ • 0.1 and Vg/v^ - 0.5. The ratio of the maxi¬
mum accelerations Sg/a^ is given by the equation
i.Zsi
•1 ri *2
and corresponds to a value of 5* The dashed line curve on the left shows the
spectrum for the primary pulse. The corresponding curve on the right is the
smma as but displaced along the frequency axis by the amount t^/tg. The
solid line, representing the spectrum for the combined pulse, is obtained by
to the ordinates of the curve the ordinates of the displaced curve
multiplied by the ratio Vg/v^.
The spectra in Fig. 2.45b are for pulses with different combinations
of tg/t& and Vg/v^, as shown in the figure. It can clearly be seen from these
2-56
plots that the secondary pulse has practically no effect on the low-frequency
region of the spectrum. This result might have been anticipated from the
material presented previously, since the maximum displacement of the ground,
which controls the maximum deformation of low-frequency systems, It practically
unaffected by the high-amplitude high-frequency acceleration pulses considered
In these exasples.
On the other hand, the high-frequency region of the spectrum is
Influenced to a very significant degree by the superimposed oscillation, since
the magnitude of the maximum input acceleration, which controls the response In
this case, is Increased significantly. In each case, the limiting value of A
becomes equal to the maximum possible value of the acceleration of the combined
pulse, ^ ♦ Sg.
In the Intermediate range of frequencies, the spectrum for the primary
pulse Is modified In two significant respects: It becomes wider, and a second
peak appears at the frequency corresponding to the peak of the displaced curve.
In addition, the peak value of the spectrum for the main pulse Is Increased, but
this change la relatively small for the range of parameters considered. From a
consideration of the manner In which the spectrum for the combined pulse la
obtained, It should be clear that, other things being equal, the distance between
peaks will Increase with decreasing value of the level of the second
peak will Increase with Increasing value of Vg/v^, and the increase In the peak
value of the spectrum for the primary pulse will be less significant for highly
peaked spectra than for spectra having a flat top.
The trend referred to in the last statement can be seen in Pig. 2.46
which Includes deformation spectra for a combination of two full-cycle sinusoidal
velocity pulses, lots that, for the same values of the parameters, the percen¬
tage increase In the peak value of the spectrum for the primary pulse is smaller
2-57
In this figure than for the spec t run given In Fig. 2.45b. The one additional
difference between the seta of curves given in the two figures Is that In
Fig. 2.46 the Halting value of A at high frequencies Is 2(a^ + a,,) Instead of
+ 8^. This difference Is a consequence of the discontinuities in the Input
acceleration function.
If the secondary component of the input function has several pulses
of nearly equal amplitudes but different durations , the spectrum for the combined
notion would exhibit several peaks corresponding to the peaks of the component
spectra.
Finally, the general shape of the curves presented In Figs. 2.45b and
2.46 emphasize that the middle region of the spectrum for a combination of pulses
cannot be determined on the basis of the maximum values alone of the Input
velocity and acceleration functions, but that proper regard should also be given
to the detailed features of these functions. In particular, the results obtained
may be quite conservative If the middle region of the spectrum Is approximated
by a horizontal line and a diagonal line of constant value of A, as was recommended
for the case of simple pulses. However, such errors are likely to be Important
only when a^/a^ is large and Vg/v^ Is small, that Is when the secondary
acceleration pulses are of very high amplitude and short duration.
2.15 Deformation Spectra for Systems Subjected to Earthquake Motions
2.15.1 General. It Is shown In this section that the significant
features of the deformation spectra corresponding to ground motions even of the
complexity of those induced by strong motion earthquakes can be estimated with
reasonable accuracy from the Information for simple pulses that has been
presented In the preceding sections. To accomplish this, the acceleration,
velocity and displacement of the ground must be known as a function of time.
2-56
The left-hand portion of the spectrum may then be estimated frcm the
characteristics of the displacement function, the middle portion may be
estimated from the characteristics of the velocity function, and the right-hand
portion may be estimated from the characteristics of the acceleration function.
The Input motions considered in this study include the nearly north-
south component of the ground motion recorded during the Eureka, California
earthquake of 21 December 1954, and the north-south component of the record
obtained during the El Centro, California earthquake of May 18, 1940. The
time histories of the acceleration, velocity, and displacement for these motions
are shown in Figs. 2.47 and 2.46. The maximum absolute value of an input func¬
tion will be Identified with the subscript o, and the subscript o,p will be
used for value of the dominant wave in the primary component of that
function. The recorded accelerograms were approximated by a series of straight
line segments, and the velocity and displacement histories were determined by
numerical Integration. The base line of the accelerograms was adjusted so that
the resulting velocity diagram oscillated about the zero line, and certain
minor adjustments were made at the beginning of the records to account for
uncertainties regarding the time of initiation of the shock.
2.15*2 Presentation of Data. The deformation spectra corresponding
to these records are given in Figs. 2.49 and 2.51 for systems with coefficients
of damping in the range between zero and 40 percent critical. In addition,
the times of occurrence of these maxima, tQ, are plotted in Figs. 2.50 and 2.52
as a function of frequency. The data used to prepare these plots are tabulated
in Appendix A along with the maximum values of the quantities u, U, x, £ and
x, and their associated times of occurrence.
Each of the curves was established with 22 data points. For the
XI Centro record, soma additional solutions ware obtained for systems with
2-59
0 * 0.02, to evaluate the detailed feature a of the apectra. These results are
presented in Fig. 2.55, la which the solutions used earlier are represented by
open circles and the additional solutions are shorn in solid circles. It can
be seen that the data points corresponding to the coarse frequency Interval
define with reasonable accuracy the salient features of the actual spectra.
The accuracy should be still better for systeas having nore than 2 percent
critical daaplng, since the irregularities of the spectra generally decrease
with increasing daaplng.
There are striking similarities between the spectra presented in
these figures and aany of the spectra for staple pulses presented earlier.
Specifically, at low frequencies, the deformation U approaches the
ground displacement, y ; at high frequencies, the acceleration A approaches
the maximal value of the ground acceleration, yQ; at the intermediate fre¬
quency range, the pseudo- velocity is nearly constant; and the maxima values of
U and A occur to the left and to the right of this Intermediate nearly flat
region. (The magnitudes of these maxima will be considered later.) Furthermore,
as mould be expected from the data and the discussion presented earlier for
staple pulse-like Inputs with continuous acceleration diagrams, the effect of
daaplng is most pronounced in the intermediate frequency range and is practically
negligible at very low and at very high frequencies.
The one major difference between the results presented in this
section and those given earlier concerns the reduction in the value of the
maximum response obtained with 2 percent critical damping. Whereas the reduc¬
tion achieved is practically negligible for the simple pulses, for the earth¬
quake motions it is quite significant, particularly in the region of the spectrum
where A attains its maximum value. This difference is due to the secondary,
high frequency component of the earthquake motion, which, because of its nearly
2-60
periodic character^ produce* an almost resonant condition. The effectiveness
of damping under such a condition is known to be great acd to increase with
Increasing duration of the excitation. The resulting reduction in response
would be expected to be particularly pronounced in the case of the SI Centro
record which is of longer duration and for which the high frequency component*
have greater amplitudes and occupy a greater portion of the record than for the
Eureka record. This prediction is substantiated by the curves presented in
Fig. 2.50 and 2.52 which show that; for systems without damping; the maximum
deformation occurs near the end of the record; while for systems with as little
as 2 percent critical damping it occurs at a much earlier tine. Vbr example;
for the El Centro record, the values of A and the associated times of occurrence
for systems with f » 20 cps are as follows:
0
sec.
A_
*
0
24.3
2.73
0.02
9.6
1.29
0.05
9.6
1.11
0.1
2.1
1.03
0.2
2.1
I.03
0.4
2.1
1.02
Because of this difference in the response of cospletely undamped
systems to the two forms of excitation; and in view of the fact that all physical
systems have some Mount of damping; the earthquake spectra for 0 ■ 0.02 will
be used as a basis of comparison; and, unless otherwise noted, they will be con¬
sidered to be comparable to the undamped spectra for alsple pulses.
In the computation of the effects of the earthquake motion*; if only
the portion of the record between the origin and time tQ had been considered;
the co^uted value of the maylmw response would obviously have been the seme.
2-61
It is important to note that, even for systems having as little as 2 percent
critical damping, the portion of the record which controls the deformation
is a small fraction of the total duration of the record.
2.15.3 Relationship Between Characteristics of Input Motions and
Response Spectra. Tor the ground record corresponding to the Eureka quake, it
can be clearly seen from Fig. 2.47 that the moat significant portion of the
motion extends from about 2.5 to 7 seconds, and thi« portion may be expected to
control the response of systems with damping. That this is indeed the case can
be seen from Fig. 2.50 which shows that, with minor exceptions, the maximum
deformation of systems with as little as 2 percent critical damping generally
occurs at less than 7 seconds.
The dominant portions of the velocity and displacement diagrams for
the Eureka shock are reproduced in Fig. 2.54a in solid lines. Superimposed on
these as dashed line curves are what are considered to be the primary components
of the motions.
In the displacement trace, the dominant pave is a skewed half-sine
pulse with an amplitude, y , slightly less than the maximum ground displacement,
o,p
yQ, and a duration of about 30 seconds, as shown in the figure. On the basis
of this information, the left-hand region of the spectrum would be expected to
be similar to the corresponding regions of the spectra shown in Fig. 2.26, with
the value of U being equal to the maximum ground displacement for frequencies
determined from the equation
l 7a
tdf < 2* y^ (»*e *»•
and with the frequency corresponding to the marimum value of U determined from
Eq. 2.48x. These frequencies are
2-62
0.076 cps
, ^ 1 1 x
*<J3‘Su?‘
and
t - (0.4) | - 0.19 cps
respectively , and agree veil with the actual data given in Fig. 2.49.
For 0 « 0.02, the absolute value of U is UQ ■ 1.J2 yQ, which considering
that y _ - 0.9 y , becomes
o,p o'
U » 1.5 y
o ' 'o,p
This value coincides with the value obtained from the design spectrum presented
in Fig. 2.26.
The primary component of the velocity trace is a full-cycle pulse
with an amplitude of about 0.7 yQ and an average duration of about 1.8 secs,
for each half-cycle. The total duration of the three major pulses in the
superimposed secondary conponent is about 1.8 secs., as shown in the figure.
The middle region of the spectrum would therefore be expected to exhibit two
major peaks. The one corresponding to the primary pulse would be expected
approximately at a frequency determined from Eq. 2.49, or at
f - 0.6/1. 8 ■ 0.33 cps,
and the second peak would be expected at a frequency
f - 0.6/0. 6 « 1 cps
These results are also in good agreement with the data given in Fig. 2.49,
where it is worth noting that the absolute maximum value of V for 0 - 0.02
occurs at 0.08 cps, and not at the frequency for which the curve for 0 ■ 0
attains its maximum value.
2-63
The magnitude of the maximum amplification factor for V in the middle
region of the spectrum is somewhat smaller than the value of 3 which one might
he temped to assume on the basis of the full-cycle velocity pulse that dominates
the ground motion. This apparent discrepancy is due to the fact that V has
been normalized with respect to yQ instead of the maximum value of the primary
component of the velocity, y - 0.7 y,.
o,p o
Because of the nearly erratic character of the ground acceleration
diagram, the magnitude of the maximum value of A cannot be estimated reliably.
However, the significant features of the high-frequency portion of the spectrum.
Including the location of the maximum value of A, can still be related to the
dominant features of the input acceleration.
For example, considering that the average duration of the most intense
pulses in the acceleration trace of the motion is of the order of 0.3 secs.,
the peak value of A would be estimated to occur at a frequency
f - 0.6/0. 3 - 2 cps.
Furthermore, since the rise time for the pulse corresponding to the maximum
input acceleration is less than one-half its duration, the frequency beyond
which A may be considered to be equal to the maximum ground acceleration is
estimated from Eq. 2.30 to be greater than
f ? 1.25/0.15 - 8.3 cps .
These results are again in good agreement with the actual datA for systems
with p - 0.02.
Referring now to the El Centro earthquake records given in Fig. 2.1*8,
one observes that the most intense waves which can be expected to control the
response are concentrated in the first 6 seconds of the acceleration and
velocity records, and in the first 10 seconds of the displacement record. The
2-61*
primary components of the waves in the early portions of the velocity and
displacement records are shown in dashed lines in Fig. 2.54b*
The dominant displacement wave is approximately a half-sine pulse
with an amplitude nearly equal to the maximum ground displacement and an
effective duration of about 6.1 seconds. Superimposed on this, there is a
secondary full-cycle wave of smaller amplitude and duration of about 2.2 sec.,
as shown in the diagram.
Considering only the contribution of the primary wave, the value of
U TOuld be expected to be equal to the maximum ground displacement for a range
of frequencies determined from Eq. 2.46, i.ew,
f < 53 S 1 ' °-04 <**
and the maximum value of U would be expected to occur (see Eq. 2.48x) at
f - (0.4) | - 0.10 cps
In addition, a second maximum, corresponding to the effect of the superimposed
full-cycle wave and of the wave preceding the primary pulse would be expected
roughly at the average frequency of these waves, or at
f - 1/2.65 - 0.38 cps
Excepting the fact that the computed value of U at f « 0.04 cps is 33 percent
greater than the estimated value of yQ, these results are in excellent agreement
with those given in Fig. 2.53.
Of the two peak values of U, the one corresponding to the lover
frequency has a magnitude of 2.06 yQ, as shown in Fig. 2.53. The difference
between this value and the value of 1.7 yQ reported earlier for a half-sine
displacement pulse is due mainly to the neglected effects of the secondary wave
and of the wave preceding the major pulse, both of which tend to increase the
2-65
response. The fact that the maximum deformation occurs near the end of the
record suggests further than the contribution of the waves following the main
pulse is not entirely negligible, although it is expected to be quite small.
The maximum possible contribution of the wave preceding the main pulse may be
considered to be approximately equal to the amplitude of the residual oscillation
induced by a full -cycle sinusoidal wave with a duration of 3.1 sec. and an
amplitude of 3 inches. For a system with a natural frequency f * 0.10 cps,
this amplitude is determined from Eq. 2.32 as 1.8 in., or 0.22 yQ.
The most significant part of the primary component of the ground
velocity is shown approximately by the dashed line in the upper diagram of
Fig. 2.54b, , It consists of a sequence of five half-cycle waves the amplitudes
and durations of which are as Indicated. The amplitudes of the major waves in
the nearly periodic, secondary component are from about 0.3 to 1.0 times the
peak amplitude of the primary components and their average period is about 0.7
seconds .
The middle region of the deformation spectrum would, therefore, be
expected to have the general appearance of the dashed-dotted line curve shown
in Fig. 2.46 with the exception that the two peaks of this curve should be
closer to each other. The peak associated with the primary waves would be
expected to occur at a frequency determined from Eq. 2.49, with the quantity
t. taken as the duration of the third wave which, because of its shape and
i.,a
amplitude, is believed to be the dominant one. This frequency is
f ? 0.6/1. 5 - 0.4 cps . (2.58)
Because of the nearly periodic character of the waves in the secondary component
of the velocity diagram, the second peak would be expected at a frequency close
to the average frequency of these waves, or at
f - ljo.1 ■ 1.4 cps .
2-66
These results are in good agreement with the exact values shown in Fig. 2.53*
The magnitude of the first peak value of V would be estimated from
the expression
V - 1.5(25.1) + 0.7 (10.6) = 1*5 in/sec. (2.59)
where the first term on the right-hand member gives the contribution of the
first four waves in accordance with Eq. 2.54, and the second term represents
a liberal estimate of the contribution of the fifth wave. The amplification
factor of 1.5 Is not appropriate for the latter wave, because its duration is
only 0.6/l. 5 times that of the most dominant pulse. The factor 0.7 was determined
from the spectrum for a versed-sine velocity pulse given in Fig. 2.17a by taking
the ordinate of the curve at a value of tjf * 0.6(0. 4) * 0.24, where 0.4 cps
represents the frequency determined in Eq. 2.56.
The value of V given in Eq. 2.59 may be expected to represent an
upper bound to the effect of the primary velocity component for systems with
0 - 0.02. Because of the nearly periodic nature of the input velocity. function,
the possible reduction due to 2 percent critical damping, although small, is
not entirely negligible in this case. On the other hand, this reduction will
be partially compensated by the Increase due to the effect of the secondary
wave. Accordingly, the estimated value should be directly comparable to the
first maximum value of V in Fig. 2.53, vhlch is
V - 3.30(13.7) - 45.2 in/sec.
Referring now to the ground acceleration diagram presented in
Fig. 2.46 it is noted that the most Intense pulses are concentrated in the
region between two and three seconds and that there are four half-cycle pulses
of nearly equal amplitude and an average duration of about 0.15 sec. each.
This information suggests that the peak value of A would be controlled by this
portion of the diagram and that it will occur approximately at a frequency
2-67
of about 0.6/0. 15 - 4 seconds. Furthermore, the presence of several high
Intensity acceleration pulses of both shorter and greater durations suggests
that the value of A would he close to its maximum value for a fairly vide range
of natural frequencies. These trends are substantiated by the actual data
presented in Figs. 2*51 and 2.55. That the response of a system having a natural
frequency of the order of 4 cps is indeed controlled by the high intensity
portion of the ground acceleration diagram can clearly be seen from the dashed
line curve in Fig. 2.53, which shows that the majority of the data points in
the frequency range between 2 and 5 seconds correspond to a value of tQ - 2.7
seconds.
Concerning the magnitude of the peak value of A, it can only be noted
that the computed value for an undamped system is about 9.5 VQ, for a system
with p - 0.02 it is 4.5 yQ, and for a system with 0 • 0.40 it is almost equal
to y . It is particularly noteworthy that the entire right-hand portion of the
spectrum for 0 ■ 0.40 is represented almost exactly by the diagonal line
A « yQ. This is also true of the corresponding spectrum for the Eureka quake
presented in Fig. 2.49.
Finally, noting that the shortest rise time for the four most Intense
pulses in the acceleration diagram of the ground is of the order of 0.05 seconds,
it is concluded that, for damped systems, the response acceleration A should
be of the order of yQ for values of f greater than 20 cps . The fact that the
value of A for f > 10 cps and 0 ■ 0.02 is almost twice as great as the maximum
ground acceleration should not be surprising, therefore.
It should perhaps be emphasised that what has been referred to as
"predicted11 or "estimated" data was arrived at after the response spectra were
evaluated. However, the degree of agreement achieved and the straight¬
forwardness of the procedure used to arrive at these results illustrate clearly
2-68
the Intimate relationship that exists between the response spectra for simple
pulses and those for complex earthquake motions, and should leave but little
doubt about the possibility of determining these spectra vlth reasonable
accuracy from the gross characteristics cf the acceleration, velocity and
displacement records of the motion.
In Figs. 2.55 and 2.56 are given response spectra for the maximum
positive and the maximum negative deformations of systems with (J » 0.02 and
P “ 0.40 for the two earthquake motions considered. In general, the two sets
of curves are in good agreement between each other. The agreement is better in
the case of the El Centro motion because the positive ahd negative parts of
the acceleration and velocity records of this motion are more nearly balanced
about the zero line than are those of the records for the Eureka earthquake.
2.16 Bpectra for Other Response quantities
2.16.1 Spectra for Relative Velocity. For the class of ground motions
considered in this study, it has been shown in Section 2.4 that the relative
velocity u due to an input velocity function y2(t) is the same as the deformation
u produced by a displacement function y^(t) of the same shape. Each response
quantity is considered to be normalized with respect to the maximum value of
the corresponding input function.
It is desirable to plot the spectral values of u on a four-way
logarithmic plot similar to that used earlier, with the vertical and the
diagonal scales representing the quantities p 0/yQ and p^O/jT , as shown
in part (b) of Fig. 2.57* The resulting spectrum for a prescribed velocity
function will then be identical to the deformation spectrum corresponding to a
displacement function of the same shape. The spectra for U can, therefore, be
constructed approximately by application of the design rules presented earlier.
2-69
In particular, the U spectrum for a half-cycle acceleration pulse may he
approximated hy the diagram given in Fig. 2.23, and the corresponding spectra
for a half-cycle velocity pulse and a half-cycle displacement pulse (i.e. full
cycle velocity pulse) may he approximated hy the diagrams given in Figs. 2.26
and 2.33, respectively.
In Figs. 2.58ataad: 2.58b the relative velocity spectra for the
earthquake records are plotted in the form described above for systems With
coefficients of damping between zero and 40 percent critical. The right-hand
diagonal scale is not shown because the values of *¥ for the input motions, are
not known. The spectra in Figs. 2. 58a and 2.38b can also be interpreted as the
deformation spectra for ground motions the velocity diagrams of which have the
same shape as the acceleration diagrams of the Eureka and the El Centro earth¬
quakes, respectively. It is of Interest to note in passing that the effect of
damping in the high-frequency regions of these spectra is considerably more
pronounced them for the corresponding deformation spectra given in Figs. 2.49
and Fig. 2.31. This difference is due to the discontinuous nature of the y*
diagrams and might have been anticipated from the data given earlier.
2.16.2 Comparison of Pseudo-Velocity and True Relative Velocity.
In the field of earthquake engineering, the relative velocity U has sometimes
used in lieu of the pseudo- velocity V. The degree of approximation involved in
replacing one quantity by the other has been investigated recently by Hudson
(Ref. 9) for three earthquake motions. The minimum value of natural frequency
considered in this study was about 0.27 cps, and the maximum value ranged from
about 3 cps to 12 cps for the three records. In general, the values of V and U
were found to be in close agreement between each other, but in some cases the
differences were of the order of 40 percent for systems with 0 - 0.20.
2-70
This problem was also investigated in the present study considering
a wider range of natural frequencies than that used before. Both pulse-like
excitations and earthquake motions were considered. Fig. 2.39 shows the results
obtained for an undamped system subjectedto a skewed versed-sine velocity pulse
with tj/t^ « l/4. For values of t^f between O.J and 0.9 the two quantities are
Identical because they both attain their maximum values during free vibration
when the system executes a simple harmonic motion and, consequently, U « pU * V.
In the low frequency region of the spectrum V is smaller than U. As t^f -vO,
the deformation U-*- yQ, the relative velocity U— -yQ, but the pseudo- velocity
V « pU -*• 0 by virtue of the fact that p -*• 0. In the high frequency region of
the spectrum, V is greater than U, the difference between the two quantities
increasing with increasing frequency.
Similar plots are given in Figs. 2.60 for systems with 2 percent
critical damping subjected to the Eureka and El Centro earthquake motions.
The striking similarities between these plots and those presented in Fig. 2.59
are further evidence of the Intimate relationship that exists between the
spectra for pulse-like excitations and earthquake motions. The agreement
between the two quantities at high frequencies is better for the El Centro
motion because the dominant waves for this motion are of shorter duration, t^,
than for the Eureka record, with the result that the frequency parameter t^f
is comparatively closer to the region where U and V may be considered to be the
same.
The effect of damping on the relationship between V and U is
illustrated in Figs. 2.6la and 2.6lb which should be self-explanatory .
2.16.3 Spectra for absolute Acceleration. The absolute acceleration
X for a system without damping is equal to the pseudo-acceleration A. Accordingly,
it may be determined directly from the right-hand diagonal scale if the
deformation spectrum.
2-71
4
As a measure of the error involved in taking X ■ A when £ / 0, in
Fig. 2.62a the response spectra for these two quantities are compared for
systems with p » 0.20 and p - 1.00, and in Fig. 2.62b the ratio A/x is plotted
as a function of frequency for different values of damping. The ground motion
in this comparison is a versed-sine velocity pulse. In Fig. 2.63 are given
similar results for systems subjected to the ground motions of the Eureka and
the El Centro earthquakes.. As before, the salient features of the curves for
the simple pulses and the earthquake motions are the same. Even for large
••
amounts of damping, the quantities X and A are very nearly the same at high
frequencies. However, in the low frequency region, the differences between
the two quantities are appreciable for large values of p.
2.16.4 Spectra for Absolute Velocity and Absolute Displacement. It
is convenient to plot the spectra for these quantities on the four-way
logarithmic grid used previously, with the diagonal and vertical scales normalized
as shown in parts (c) and (d) of Fig. 2.37* In this figure the quantities
[/y(,)i i "* [/</ y(,)4T)iti represent, respectively, the maximum values
of the first and the second integrals of the ground displacement function.
By tirtue of Eq. 2.11, the spectrum of X corresponding to a ground
velocity function can also be Interpreted as the acceleration spectrum for an
input acceleration having the shape of the prescribed velocity function.
Similarly, the spectrum of X for a given displacement function can he interpreted
as that of X due to an input acceleration of the shape of the prescribed
displacement function. Recalling now that, for lightly damped systems, X may
be replaced by the pseudo- acceleration A, it is concluded that the velocity
spectrum X for a prescribed velocity disturbance may be considered to be the
same as the deformation spectrum for an acceleration disturbance of the same
shape. A similar statement cafe also be made for the displacement spectrum X.
2-72
The deformation spectra can, of course, he approximated by application of the
appropriate design rules presented in this report. It must be emphasized that
this approach is valid only for lightly damped systems.
As an illustration, in Fig. 2.64 are given velocity spectra, plotted
in the form described above, for systems subjected to the full-cycle velocity
pulse considered previously in Fig. 2.37* As would be expected from the
preceding discussion, the spectrum for £ * 0 is similar to the deformation
spectrum corresponding to a full -cycle acceleration pulse (half-cycle
displacement pulse), such as that considered in Fig. 2.34. It is important to
note that the trends of corresponding curves in Figs. 2.34 and 2.64, while
similar to each other for medium- frequency and high-frequency systems, differ
significantly for low-frequency systems with values of 0 on the order of 0.2
or more. These differences cure analogous to those between the quantities A
and X considered in Figs. 2.62.
In Figs. 2.63a and 2.63b are given Bimilar spectra for the Eureka
and El Centro Earthquake records. For values of 0 less than about 0.10, these
spectra can also be interpreted as deformation spectra for ground motions the
acceleration diagrams of which have the shapes of the velocity diagrams of
the earthquake motions considered.
2-73
SECTION 3
RESPONSE OF INELASTIC SYSTEMS
3.1 General
This chapter is concerned with the response of inelastic systems
having a single degree of freedom. Primary attention is given to elastoplastic
systems and, in an exploratory way, to bilinear systems of the softening type.
Only the max liman deformations of the systems are investigated.
Figure 3.1a shows the resistance-deformation relationship for a
bilinear system. The symbols and kg denote the slopes of the first and
second portions of the diagram as indicated. For a bilinear system of the
softening type, k^ < k^, and for an elastoplastic system, kg » 0. The yield
levels in the two directions of deformation are considered to be the same, and
unloading from a point of maximum deformation is assumed to take place along a
line parallel to the initial elastic portion of the curve. A typical cycle of
loading, unloading and reloading is shown in the figure. The yield point
deformation is denoted by u , and the absolute maximum deformation, without
y
regards to sign, is denoted by uffl. In an analogous manner, the yield point
resistance is designated by Q^, and the maximum spring force by For an
elastoplastic system, the force Q^is, of course, equal to for deformations
in excess of the yield point deformation.
The ground motions considered Include five pulse-like excitations and
the two earthquake records used in the study of elastic systems. In addition,
the effects of certain limiting forms of excitation are studied. The
acceleration, velocity and displacement diagrams for the simple pulses are
shown in Fig. 3.2. The acceleration diagrams consist of straight line segments
and, except for one pulse, they are discontinuous at the beginning and the end
5-1
of the diagram. The velocity diagrams have from one to four parabolic half¬
cycles of oscillation. Hie initial values of the velocity and displacement
diagrams axe zero in all cues.
3.2 Definitions and Fundamental Relations
It is convenient and instructive to relate the maximum response of
the inelastic system to that of an elastic system having the same stiffness as
the initial stiffness of the inelastic system. Let uq be the absolute* value
of the maximum deformation of the associated elastic system, and be the
corresponding spring force, as shown in Fig. 5. lb. The yield resistance of the
inelastic system, Q , may then be expressed as a fraction of the resistance Q
y o
required for elastic behavior. The ratio which is also equal to u y/uQ,
will be referred to as the reduction factor and will be denoted by the symbol c.
That is,
Q u
(3-1)
Q u
c • -X ■ -X
Q u
o o
For an elastic system, the quantities and u^ may be considered to be equal
to and uq, respectively. Accordingly, the reduction factor is equal to
unity in this case. For a system that deforms in the inelastic range, c is
evidently smaller than unity.
The reciprocal of the reduction factor, l/c, expresses the intensity
of the ground motion in terms of that which the system can withstand elastically,
and will be referred to as the overload factor.
The maximum deformation of the inelastic system, u , can conveniently
m
be expressed in terms of its yield point deformation, u . The dimensionless
ratio
(3-2)
This notation is not consistent with that used in Section 2, where the subscript
o referred to the MV^"1"11 value of the quantity taken with its appropriate sign.
3-2
will be referred to as the ductility factor. With this notation, the maximal
Inelastic deformation of the system is (u - l)u .
y
With the values of c and u known, the ratios UB/U0 and Qb/Q(> can be
determined from the following equations
~ - u c
o
and
(3.3)
(3.M
Equation 3*3 follows directly from Eqs. 3*1 and 3*2, whereas Eq. J.h can
readily be derived by reference to Fig. 3.1b.
For a system without damping, the spring force is proportional to
the acceleration of the mass, and consequently
v*.
w
The symbols x and x denote the absolute maximum
mo
and the elastic systems, respectively.
(3.5)
accelerations of the inelastic
3.3 Response to Limiting Forms of Ground Excitation
With a view of establishing certain guide lines for the interpretation
of the results to be presented later, we consider first the relationships
between the deformation of the elastoplastic system and the associated
elastic system for certain limiting forms of ground excitation. These include
(a) an Instantaneous displacement change,
(b) an instantaneous velocity change, and
(e) an instantaneous acceleration change.
The system is presumed to be undamped and initially at rest.
3-3
3 -5*1 Instantaneous Displacement Change. For a system subjected to
an Instantaneous displacement change, -yQ, the "initial" value of the resulting
deformation will be y , irrespective of whether the system behaves elastically
or deforms in the plastic range. Furthermore, since there is no additional
energy Imparted to the system after the displacement change has taken place,
the extremum values of deformation for the ensuing motion will be numerically
equal to or less than yQ, and the initial deformation will be the absolute
maximum deformation. In other words.
u
u ■ y
o 'o
and the reduction factor for the inelastic system, determined from Eq. 3*3> is
1
c ■ —
h
(3.6)
Note that this expression is independent of the ratio kg/k^.
As an illustration. Fig. 3.1c shows the resistance-deformation
diagram of an elastoplastlc system with uy < yQ. The abscissas of points b
and c define, respectively, the deformations of the inelastic and the associated
elastic systems immediately after the initial displacement change. The
extremum values of deformation for the ensuing motion will correspond to points
c and c' of this figure for the elastic system, and to points b and b' for the
elastoplastlc system.
Although the initial deformations for the inelastic and elastic
systems are the same under the conditions assumed, the energies imparted to
these two systems are different. The energy imparted to an elastic system is
E - i Q u ,
2 o o'
(3-7*)
and that Imparted to an inelastic system is given by the equation
h ■ Vu» * I V + § k2(u. - V2 •
3-^
The latter equation can also be written In the form
QyUy [(2M - 1) + ^ (M - l)2]
Utilising the fact that when ■ uQ, 0y/<4o ■ u y/ue • l/n>
following expreselon for ratio of the two energies.
(3*7b)
one obtain* the
(2U - 1) + ~ (u - l)2
_ _ 1 _
(3.8)
It way be noted in passing that for an elastoplastlc system with u * 5 the
ratio X ^/Eo ■ O.36.
3*3*2 Instantaneous Velocity Change. The energy Imparted to a
1 2
system by an Instantaneous velocity change, vq, is ^ mv^ , irrespective of
whether the system remains elastic or not. Consequently, the energies absorbed
by an elastic and an inelastic system up to the point of their respective
maximum deformation will also be the same. This equality is expressed by the
equation
\ *1 uo " ? V 'V [(* - X) * ^ ’ 1)2] (5-9)
whence
uQ2 - uy2 [(2u - 1) + ^ (m - l)2]
and the reduction factor becomes
c - ■ = = ■■■■■ ■ . , (3*10)
J(2n - l) + ^ (u - l)2
Consider now an elastic and an inelastic system subjected to a
prescribed motion of arbitrary shape, but assume that the conditions are such
that (a) the absolute maximal deformations of both systems occur during free
vibration and (b) no yielding occurs during forced vibration.
3-5
By virtue of the second restriction, the energy of the two systems
at the beginning of free vibration will be the same, and, from the material
Just presented, It follovs that the maximum deformations of these systems will
also be governed by Eq. 3.9 , with the reduction factor c given by Eq. 3. 10.
3.3.3 Instantaneous Acceleration Change. The effect of a ground
acceleration, y(t), can most conveniently be analyzed by considering the
equivalent problem of a force -my( t) applied to a fixed-base structure.
For an acceleration step of infinite duration, yQ, the work performed
by the external force up to the point of maximum deformation is [ -m y uj » and
the energy absorbed by the structure is given by the right-hand member of
Eq. 3.9* By equating these two quantities, one obtains
_ 2u
(2n - 1) +
(u - l)2
(3.11m)
where p - Vk^/m . For an elastic system, uy - uo and u » 1, and Eq. 3.11a
reduces to
P2u
- 1_£ - 2 (3.11*)
#•
yo
The reduction factor c is obtained as the ratio of Eqs. 3-H* and
3.11b, yielding
c
_ I*
(2n - 1) ♦
(m - 1 f
(3.12)
In this case, it is instructive to consider also the ratio of the
forces developed in the inelastic and the associated elastic systems.
From Eqs. 3.4 and 3.12, one obtains
3-6
(5.13)
«5_ M[1 * A(“'1)]
Q° (2m - 1) + ^ (n - l)2
*1
Tide ratio is evaluated in the following table for several values of kg/k^ and |i.
Values of Q^/Qq as Given by Eq. 3 *13
u
“A ■
0 kj,/^ . 0.1 - 0.2
*2/*! - 0.5
k2/kx « 1.0
1
1.0
1.0 1.0
1.0
1.0
2
0.67
0.71 0.75 •
0.94
1.0
5
0.56
0.66 0.74
0.88
1.0
10
0.53
0.70 0.80
0.92
1.0
00
0.50
1.0 1.0
1.0
1.0
It can be seen that the possible range of variation of Q^/Qo is from 1.0 to
0,5.
3.3.4 Discussion. The results presented in the preceding para-
graphs can be summarized as follows:
(a) For a system subjected to an Instantaneous displacement change,
the maximum deformations of the elastic and inelastic systems are the same.
(b) For a system subjected to an instantaneous velocity change, the
energy absorbed by the system up to the point of maximum deformation for the
elastic case is the same as that for the inelastic case, and the reduction
factor is given by Eq. 3. 10. This relationship is also valid for an arbitrary
ground motion, provided the elastic and the inelastic systems both reach their
absolute maximum deformation during free vibration, and the inelastic system
behaves elastically during forced vibration. The latter condition requires
that the yield level of the system be equal to or greater than the maximum
deformation attained by the elastic system during forced vibration.
3-7
(c) For a system subjected to an instantaneous acceleration change,
the spring force for the inelastic system can be no less than 50 percent
of that for the associated elastic system, the actual magnitude of the reduction
being a function of the ratio and of the amount of inelastic deformation
that can be tolerated (See Eq. 3*13)*
It would be expected that the first relationship involving con¬
servation of maximum deformations would also be applicable to systems subjected
to ground displacements for which the rise time is small in comparison to the
natural period of the system. The second relationship, involving conservation
of energies, would be expected to apply also to quarter-cycle velocity pulses
for which the rise time is small in comparison to the natural period of the
system (i.e., half -cycle acceleration pulses of short duration), and, possibly,
to half -cycle velocity pulses of short rise time and sufficiently long decay
time such that, at the time of the first maxim* deformation, the value of
the ground velocity is close to its maxi mm value. Finally, the third relation¬
ship would also be expected to be valid for acceleration pulses with a sharp
rise — * long duration in comparison to the natural period of the system.
In the following table are listed the values of the reduction factor,
c, and of the ratio ^m/uc corresponding to different values of the ductility
ratio for the three limiting forms of excitation investigated. The systems
considered are of the elastoplastic type, i.e. ^/^l *
Reduction Factor, c -
a /q - u /u
*y/ 0 y' 0
Values of UB/U0
m ||C
u
Displac .
Change,
Eq. 3*6
Velocity
Change,
Eq. 3-10
Acceler.
Change,
Eq. 3.12
Displac.
Change
Velocity
Change
Acceler.
Change
1
1.25
1.5
2
3
5
10
1.00
0.80
0.6?
0.50
0.33
0.20
0.10
1.00
0.82
0.71
0.58
0.*5
0.33
0.23
1.00
0.83
0.75
0.67
0.60
0.56
0.53
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.02
l.p6
1.16
1.3*
1.67
2.29
1.00
1.04
1.12
1.33
1.80
2.78
5.26
3-8
It can be seen from this table that even a relatively small amount of Inelastic
deformation, as represented by a value of u on the order of 1.5, produces a
significant reduction in the value of the required yield resistance and a
relatively small increase in the value of the maximum deformation. Considering
that for values of u less than 2 the values of the reduction factor and of the
ratio um/u0 for the three forms of excitation differ by less than 55 percent,
it is concluded that the form of excitation is not a very significant parameter
as long as the magnitude of inelastic deformation involved is small. However,
for the greater values of u, the differences between the three sets of results
are quite important, especially when the effects of a displacement change and
an acceleration change are compared. The results for a velocity change are
intermediate between those for a displacement change and an acceleration change.
5.^ Relations Between Response of Elastic and Inelastic Systems
In Fig. 5.5 the response spectrum for the absolute maximum deformation
u^ of elastoplaatlc systems with c * 0.25 is compared with the spectrum for the
corresponding deformation uq of the associated elastic systems. The inelastic
spectrum is applicable to systems for which the yield level is one fourth of
that required for elastic behavior, or alternatively, to ground motions that are
four times as Intense as those which the systems can withstand elastically. The
ground motion is the parabolic velocity pulse considered earlier in Figs. 2.19
and 2.20. The systems are considered to have no damping and to be initially at
rest. The natural frequency of the inelastic system is determined from the slope
of the initial elastic portion of the resistance-deformation diagram.
It can be seen from this figure that the same percentage reduction in
the yield level of the system (or equivalently, the same overload) has quite
different effects on systems with different natural frequencies. For flexible
systems, u and u are equal; for high-frequency systems, u is significantly
greater than uQ; and for medium- frequency systems, u^ is smaller than uQ.
3-9
Figures 3*4 show the effect of progressively reducing the yield
level of the elastoplastic system, lhe reduction factor, c, is plotted as a
function of the ductility ratio, u, for fixed values of the frequency parameter.
It is noted that for values of t^f <0.2, the reduction factor is
represented almost exactly "by the expression l/y (Eq. 3.6); in other words, the
maximum deformations of the inelastic and elastic systems are the same,
irrespective of the yield level involved.
For values of t^f > 1, the reduction factors are generally greater
than those obtained by the relation 1/ u; furthermore, they are quite sensitive
to variations in the value of t^f, as can readily be seen from Fig. 3.4b. As
t^f approaches Infinity, the input function approaches an acceleration step of
long duration and, as would be expected from the discussion in Section 3.3,
the results approach the relation c - u/(2 u - l), which is a specialized form
of Eq. 3.12.
For the particular case investigated, the expression c « l//2u - 1,
which defines the effect of an instantaneous velocity change, may be considered
to be valid for a value of t^f of about 1.23. It can be shown that for this
value of t^f, the ratio V/yQ for the associated elastic system is slightly
smaller than unity.
Figure 3* 4c, which refers to a value of t^f ■ 0.5# is typical of the
results obtained for medium- frequency systems. In this case, the absolute
mnHnmm deformation of the elastic system, uQ, occurs during free vibration
and corresponds to the second extremum value, while the first extremum occurs
during forced vibration and is equal to 0.37 uq. The curve abc in this figure
refers to the first extremum, and the curve de refers to the second. For
values of 0.57uq < uy < uq, yielding initiates after termination of the pulse,
and, as would be anticipated from the material presented in Section 3*5*2, the
equation of line ab is c ■ l//2u - 1.
3-10
We digress now from the discussion of Fig. 5. 4c and refer to
Fig. 3.5 to define the conditions under vhich the expression c - l//2|i - 1 is
applicable. The dashed curve in this figure gives the marl man deformation of
an elastic system during forced vibration, and the solid curve, discontinued
in the regions vhere it lies below the dashed curve, gives the corresponding
deformation during free vibration. It follows that the equation c - l/72ji - 1
is valid in the regions of the spectrum vhere the absolute maxi mas deformation
occurs during free vibration, provided the yield level of the system u is
y
greater than the value of deformation obtained from the dashed curve. For
example, for t^f * 0.68, vhere the difference between the ordinates of the two
curves is greatest, the expression l//2 y - 1 will be valid for values of
From the table given on p. it can be seen that these values of u y/uQ correspond
to values of u < 2.3.
Returning now to Fig. 3.4c, ve observe that as uy is decreased below
0.37 uq, yielding initiates during forced Vibration, and the expression
c « 1//ST7T no longer applicable. In fact, the magnitude of the second
extremum decreases sharply, as can be seen from the break of the curve abc at
point b. However, for values of u^, between those corresponding to points d and
f , the absolute
deformation still corresponds to the second ext:
Finally, as uy approaches its limiting value of zero, the value of the
deformation will approach the maximal displacement of the ground, yQ.
It should be noted that there can be more than one yield level
corresponding to a given value of y. Fig. 3.4c shews three yield levels
corresponding to a value of y >2. Although all three of these solutions are
distinct and "stable", from a design standpoint it is desirable to consider
3-11
only the one corresponding to the highest yield level. This amounts to
replacing the portion of be vhlch curves to the left by the vertical shown
dotted.
Figures 3.6a and 3.6b show the relationship between the reduction
factor and the ductility factor for elastoplastic systems subjected to parabolic
velocity pulses with one and two cycles of oscillation, as shown in the inset
diagrams. In each case, the frequency parameter t^f » 0.50.
For the conditions considered In Fig. 3.6a, the iMAx-iimim deformation
of the elastic system, uq, corresponds to the third extremum, and, as a
consequence, the resulting plot consists of three branches with two transition
curves. As an Illustration of the significance of the various branches, ve
note that, for values of O.O56 u < u < 0.28 u , yielding initiates at the
o y o .
instant that the associated elastic system attains its first extremum, but
that the absolute maximum deformation is obtained at the second extremum instead
of at the first. Under the conditions considered in Fig. 3.6b, the peak
deformation uq corresponds to the fifth extremum, and, consequently, the graph
consists of five branches with four transition curves.
It must be emphasized that the discontinuities in the plots presented
in Figs. 3»4c and 3*6 for medium- frequency systems do not, in general, occur
for systems with low and high frequencies, because the order of the controlling
maximum in the latter cases is usually the same for the elastic and the
inelastic systems.
In Figs. 3.7a through 3>dc is presented information on the response
of elastoplastic systems subjected to the two earthquake motions considered in
the study of elastic systems. The plots in Figs. 3*7 are analogous to those
given in Fig. 3.3, and the plots in Figs. 3.8 are analogous to those given in
Figs. 3.4 and 3.6. The similarities between these curves and the corresponding
curves for the simple pulses are indeed most Impressive.
3-12
In Fig. 3.9 the spectra of maxi man deformation for elastoplastic
and elastic systems subjected to the Eureka earthquake are compared vith those
for bilinear systems with k^/k^ «0.2. The yield level of each inelastic
system is considered to be one fourth of that required for elastic behavior.
It can be seen that, in the low-frequency region of the spectrum, the maximum
deformations for all systems are for all practical purposes identical. In the
medium- frequency region, the differences in the results are greater but still
insignificant. The major differences occur in the high-frequency region, where
the results appear to be quite sensitive to the ratio of kg/k^. These general
trends are in agreement with those obtained on the basis of the limiting forms
of excitation considered in Section 3.3.
The effect of the parameter k^/k^ on the maximum response of the
system is illustrated in Fig. 3. 10. Results are presented for selected natural
frequencies and yield levels. The systems are assumed to have a damping factor
of 2 percent. It can be seen that the effect of k^/k^ is Important only for
high-frequency systems.
3.5 Deformation Spectra for Elastoplastic Systems
3.3.I General. The design of an elastoplastic system Involves
essentially the determination of the yield strength (or yield point deformation)
necessary to limit the maximum Inelastic deformation of the system to a
prescribed value. It is desirable, therefore, to define the deformation spectra
for Inelastic systems in such a manner that this information can readily be
determined.
With this in mind, the relative pseudo- velocity for an elastoplastic
system la defined, as In Ref. 10, by the expression
V - puy (5.14)
5-15
where p denotes the undamped circular natural frequency of the system
corresponding to the Initial elastic range of its load-deformation diagram,
and u is the yield point deformation, (not the max liman deformation). Thus,
*
the quantity
represents the max inn an strain energy that the system must be capable of de¬
veloping without yielding. The relative pseudo -velocity spectrum Is considered
to be a plot of V against frequency for fixed values of the ductility ratio, (i.
The pseudo-acceleration A, is defined as
Qy - c w (3.16)
where the lateral force coefficient, C, as In the case of an elastic system,
is equal to the value of A expressed in units of gravity. For an elastoplastlc
system without damping, Eq. 3.15 Is also equal to the maximum value of the
acceleration of the mass. It should be noted that these definitions of V and
A for the Inelastic system are consistent with those used for the elastic
system, since, as previously noted, the yield deformation of the elastic system
may be considered to be equal to Its maximum defoimatlon.
On a logarithmic plot of V against frequency similar to that used
for elastic systems, the set of diagonal lines extending In the north-east
direction represents values of constant yield deformation, u^, and the set of
lines extending In the north-west direction represents values of constant pseudo-
acceleration. Thus the values of u^, V and A corresponding to a prescribed
ductility ratio can be read directly. The value of the maximum
then be determined from the expression u^ ■ 4 uy.
deformation can
3.5*2 Spectra for a Half -Cycle Acceleration Pulse. In Fig. 3. 11
are given deformation spectra, as defined above, for undamped systems with
ductility ratios In the range between 1 (elastic case) and 10. Hie ground
motion Is an Initially peaked triangular acceleration pulse. As in previous
plots, the spectral quantities are normalized with respect to the value
of the corresponding ground motion. Note that the ratio of the ordinates of
the curves for an inelastic and elastic system, represents the reduction factor,
c, for the particular condition considered.
In the low>frequency region of the spectrum, the relationship between
the curves for the inelastic and the elastic systems is represented almost
exactly by the expression c » l//2u - 1 obtained from Eq. 3*10. This result
was anticipated In Section 3.3.^, since as t.f approaches zero, the ground
velocity approaches a step function for which Eq. 3. 10 applies exactly. In a
similar manner, the limiting values of the pseudo-acceleration A at high
frequencies are as defined by the right-hand member of Eq. 3.11a. The
transition curves between these limiting values are smooth in this case, because
the absolute «■ deformations lb r both the elastic and the Inelastic systems
correspond to the first extremum.
3.5.3 8pectra for Half -Cycle Velocity and Displacement Pulses. In
Figs. 3.12a through 3*l4b are given deformation spectra for elastoplastlc
systems with zero and 10 percent critical damping subjected to three different
forms of ground motion, as shown In the inset diagrams.
The data used to prepare these plots were obtained on the ILLIAC,
the digital coaqmter of the University of Illinois as follows. First, the
— xlmm deformation of the elastic system was computed for selected values of
the frequency parameter. Then, the maximum response of the inelastic systems
waa evaluated for a range of yield levels, and the results for each value of
5-15
the frequency parameter were plotted in the form presented In Figs* 2.4 and
2.6. The values of u corresponding to the selected ductility ratios were
¥
finally determined from these plots*
The salient features of these curves are as follows:
(a) At low frequencies, the relationship c ® l/u is applicable to
all cases considered, with the result that both u and u are equal to the
o m
maximum ground displacement, y . Furthermore, as the yield level of the system
Is decreased (or as u Increases), the expression u^ = yQ is valid for a wider
range of natural frequencies than for the corresponding elastic system. This
trend Is particularly noticeable In the case of the half-cycle displacement
pulses considered In Figs. 2.14. Note that the break in the left-hand portion
of the curves in this figure shifts to the right with Increasing values of u>
(b) At high frequencies, the limiting values of A are essentially
as defined by Bq. 3.11a* for the ground motions which have discontinuous
accelerations. On the other hand, for the continuous acceleration functions,
the limiting value of A may be considered to be the same for both the elastic
and the Inelastic systems.
(c) In the Intermediate range of frequencies, the relationship
between the' elastic and inelastic systems is In general coaplex. It can
broadly be said, however, that the reduction factors corresponding to a half¬
cycle displacement pulse are greater than those for a half -cycle velocity pulse.
Design Rules. For design purposes, the relationship between the
deformation spectra for elastoplastlc and elastic systems nay be expressed
approximately as follows:
(l) For the low-frequency range of the spectrum for which the
—tHmiwi deformation of an elastic system may be considered to be equal to the
maximum ground displacement, yQ, the maximum deformations of the inelastic and
^Equation 2.11a is strictly applicable to an undamped system only.
3-16
elastic systeas are for all practical purposes the sums, i.e. , u - u » y .
Consequently, the reduction factor Is given by the expression c - l/u.
(2) For the Intermediate range of frequencies, this following
relationships are applicable:
(a) When UQ Is equal to or smaller than the navi mum Input dis¬
placement and VQ is of the order of 1.5 times the maximum Input velocity, as
is the case with half -cycle velocity pulses, the maximum deformation of the
Inelastic and the elastic systems may be considered to be the same up to a
value of t .f slightly greater than the value corresponding to the peak of the
elastic spectrum. In this case, the reduction factor is again given by the
expression c « l/u.
(b) On the other hand, when UQ is greater than yQ, and Vq is
of the order of 2 to 3 times yQ, as is the case with half-cycle displacement
pulses, the maximum deformation of the Inelastic system Is generally less than
that of the corresponding elastic system. In general, the greater amplification
factors of Uq and VQ for the elastic spectrum, the more conservative are the
results obtained by application of the relation c = l/u. The differences are
particularly noticeable for the larger values of u« However, If the degree of
conservatism Implied by the use of this relationship can be tolerated, then
the expression c * l/u can be considered to be valid up to a value of t^f
located approximately at one-third the distance between the value of t^f
corresponding to the peak of the elastic spectrum ana the value of t^f beyond
the peak for which the amplification factor for V is one.
(5) For the high-frequency range of the spectrum where the pseudo-
acceleration A may be considered to be constant, the relationship between the
inelastic and the elastic spectra may be stated in terms of the magnitude of
the amplification factor for the elastic system. When the amplification factor
5-17
Is of the order of two, as would be the case for an input acceleration with a
discontinuity equal to the maximum input acceleration, the reduction factor may
be approximated by the expression c » u/(2u - 1). On the other hand, if the
amplification factor of A for the elastic system is one, as would be the case
for an acceleration function without any discontinuities, the reduction factor
may be considered to be unity. In other words, the ™**<*nm forces for the
elastoplastic and the elastic systems may be considered to be equal.
(4) For the range of frequencies betweeD those covered under (2) and
(3)» the reduction factor is sensitive to changes in the value of the natural
frequency. However, the spectrum curves for this range can usually be deter¬
mined by drawing smooth transition curves between the curves applicable to the
ranges considered in items (2) and (3). When this cannot be done readily, the
equations
nay be used as guide posts. The first equation may be considered to be valid
at a frequency for which the relative pseudo-velocity of the associated elastic
system is from shout 1.0 to 0.8 times the maximum ground velocity, and the
second equation may be considered to be valid at a frequency for which the
pseudo-acceleration of the associated elastic system is of the order of 2.
It follows that the response spectrum for an elastoplastic system
corresponding to a specified value of the ductility ratio can be obtained from
the spectrum applicable to the associated elastic system simply by dividing the
ordinates of the elastic spectrum by a factor which depends on the value of the
ductility ratio but which is different for tbs different frequency ranges.
For convenience of reference, these factors are suamarlsed in Fig. 3.15 for a
representative spectrum corresponding to a half-cycle pulse of ground velocity.
3-18
In the application of these rules to design, it is suggested that the spectrum
for the elastic system he represented by a smooth curve, without the undulations
that are characteristic of response spectra.
These rules are proposed for systems with moderate amounts of damping
(of the order of 10 percent critical or less) and ground motions for which the
primary or dominant component may be represented either by a half-cycle velocity
pulse or by a half -cycle displacement pulse. The reader is cautioned against
using these rules for displacement pulses with two or more half -cycles of nearly
equal amplitudes and durations.
Relative Effects of Damping and Inelastic Action. On coeiparlng the
inelastic spectra presented in this section with the corresponding spectra for
damped elastic systems given in Section 2, it can be seen that the relative
effects of damping and inelastic action in reducing the magnitude of the
required resistance are quite different in the various regions of the spectrum.
In particular, in the low-frequency range of the spectrum for which the effect
of damping may be considered to be negligible, the effect of inelastic action
is extremely Important. These results show clearly that, in general, the effect
of inelastic action cannot be considered in terms of a fixed amount of
"equivalent damping".
5*5*^ Spectra for Multiple -Cycle Velocity Pulses. For half-cycle
displacement pulses, it has been noted that in the regions of the spectrum
where U and V attain their maximum values, the absolute maximum deformation of
the elastoplastlc system is generally smaller than that for the corresponding
elastic system and that the results obtained from the expression c - l/u may
be fairly conservative for large values of u.
This effect is exaggerated under more nearly periodic excitations,
as can be seen from the spectra presented in Figs. 5.16 and 5.17. These spectra
5-19
are for velocity functions composed of three and four parabolic half-cycles,
respectively. The details of the input functions are shown in the inset
diagrams. Only the middle regions of the spectra are presented, since at the
regions of low and high frequency the results are identical to those presented
in Figs. 3.14.
It is of some interest to note that, if only the peak values of the
curves corresponding to the Inelastic and elastic curves are compared, the
expression l/u leads to reasonably accurate results.
3.5.5 Spectra for Earthquake Motions. The deformation spectra
presented in Figs. J.lQ and 3*19 are self-explanatory. They refer to systems
with 2 percent critical damping subjected to the Eureka and El Centro earth¬
quake records considered earlier, and are directly comparable to the corres¬
ponding spectra presented for simple ground motions. It is important to note
that, even for these complex input motions, the average relationships between
the Inelastic spectra and the corresponding elastic spectra for the various
ranges of frequency are in very good agreement with the approximate rules
presented for pulse -type of excitations.
3-20
REFERENCES
1. Sauer, Fred M. , "Ground Motion Produced by Aboveground Nuclear
Explosions", SRI Project, AFSWC-TR- 59-71, April 1959, (Secret).
2. Parsons, R. M. Company, "A Guide for the Design of Shock Isolation
Systems for Underground Protective Structures", Air Force Special
Weapons Center, AFSWC-TDR-62-64, December 1962, pp. 2-24 to 2-31.
3. Jacobsen, L. S. and Ayre, R. S., "Engineering Vibrations", McGraw-Hill,
New York, 1958*
4. Harris, C. M. and Crede, C. E., "Shock and Vibration Handbook", Three
Volumes, McGraw-Hill, 1961.
5. Barton, M. V., Chobotov, V. and Fung, Y. C., "A Collection of Information
on Shock Spectrum of a Linear System", Space Technology Laboratories,
Inc., Engineering Mechanics Report EM 11-9, July 1961.
6. Fung, Y. C., "On the Response Spectrum of Low Frequency Mass-Spring
Systems Subjected to Ground Shock", Space Technology Laboratories, Inc.,
Engineering Mechanics Report EM 11-5, April 1961.
7. Fung, Y. C., "Shock Loading and Response Spectra", Colloquim on Shock
and Structural Response, ASME, November i960, pp. 1-17 .
6. Newmark, N. M. and Hall, W. J., "Preliminary Design Methods for Underground
Protective Structures", Air Force Special Weapons Center Report,
AFSWC-TDR-62-6, June 1962 (Secret), Chapter 7.
9. Hudson, D. E., "Some Problems in the Application of Spectrum Techniques
to Strong Motion Earthquake Analysis", Bulletin of the Selsmo logical
Society of America, Vol. 52, pp. 417-430, April 1962.
10. Veletsos, A. S. and Newmark, N. M., "Effect of Inelastic Behavior on
the Response of Simple Systems to Earthquake Motions", Proceedings of
the Second World Conference on Earthquake Engineering, Japan, p. 895, i960.
11. Newmark, N. M. , "A Method of Computation for Structural Dynamics",
Transactions of ASCE, Vol. 127, 1962, Part I, p. 1406.
12. Chelapatl, C. V. and Ihrig, Ann H., "Computation of Dynamic Response of
Inelastic Shear-Beam Systems", Structural Research Program 1719,
Dept, of Civil Engineering, University of Illinois, 1961.
4-1
TABLE 1
COORDINATES OF MAXIMUM VALUE OF A FOR HALF -CYCLE ACCELERATION PULSES
Pulse
Type
*r,a
*d
Va
Coordinates of Max.
=? V
Values
A
r
• •
of A
V
Triangular
1
l
1.26
0.67
1.26
0.67
Triangular
0.5
0.5
1.57
0.95
1.45
0.75
Versed Sine
0.5
0.52
1.72
1.0
1.63
0.8
Half -sine
0.5
O.32
1.77
0.85
1.75
0.7
Sieved
Versed
0.25
0.16
1.83
l.h
1.59
0.8
Sine
0.125
0.08
1.90
2.0
1.55
0.8
Triangular
0
0
2.00
00
1.26
0.67
Rectangular
0
0
2.00
0.5 or
greater
2.00
0.5
5-1
TABLE 2
COORDINATES OF MAXIMUM VALUE OF V FOR HALF-CYCIE VELOCITY PULSES
Results may not be accurate to the number of significant figures recorded
Pulse Acceler.
No. Pulse
1
\ _
2a
2b
2c
2d
q=x-
5a
5b
5c
4a
4b
A=^
5a
5b
6a
7a
7b
7c
6
Velocity
Pulse
I
*d
Li
*a
Coordinates of Max.
Value of V
V
0
e
yo
V
t f
o,v
O
0
1.0
—
—
O
0
1.26
0.66
0.66
0.125
0.125
1.55
0.67
0.67
0.25
0.25
1.40
0.75
0.75
0.5
0.5
1.45
0.75
0.75
0.125
0.08
1.54
0.84
0.84
0.25
0.16
1.60
0.84
0.84
0.5
0.52
1.65
0.84
0.84
0.125
1.44
1.9
0.84
0.25
1.55
1.1
0.84
0.555
0.21
1.70
0.67
O.85
0.5
0.52
1.72
0.68
0.86
0.5
0.52
1.5*
0.95
O.69
0.5
0.25
0.5
0.25
1.72
0.85
0.85
0.5
0.25
1.75
0.67
0.89
0
0
2.0
0.50
1.00
5.2
6-1
(a) Half-Cycle Velocity Pulse.
(b) Displacement Pulse With
Partial Recovery.
(c) Half-Cycle Displacement (d) Full -Cycle Displacement
Pulse. Pulse.
PIG. 2.2 GROUND MOTIONS OF INTEREST
6-2
6-3
FIG. 2.4 SPECTRA FOR MAXIMUM AND MINIMUM ACCELERATIONS OF THE MASS
Undaoped Elastic Systems Subjected to a Half -Sine Acceleration Pulse
% uoi|oj©|90ov punoi© uinuiiXDM %
* «8Dfl >0 U0UDJ9|900V * X
6-5
uoj jDJ9|Q00y punoiQ mnmixp^i _ %
880^| JO UO!4OJ9|0OOy H
6-6 ■
20
6-7
2.7 SPECTRA FOR ABSOLUTE MAXIMUM AND ABSOLUTE MINIMUM ACCELERATIONS OF THE MASS
Undamped Elastic Systems Subjected to a Versed-Sine Acceleration Pulse
6-8
FIG. 2.8 DEFORMATION SPECTRUM FOR UNDAMPED ELASTIC SYSTEMS
SUBJECTED TO A HALF-SINE ACCELERATION PULSE
6-9
2.9 DEFORMATION SPECTRA FOR UNDAMPED ELASTIC SYSTEMS SUBJECTED
TO SKEWED VERSED-SINE ACCELERATION PULSES
( 0 1 d o s Bon)
A *Al!00|9A-opnesd
6-11
FIG. 2.11 DESIGN SPECTRUM FOR THE ABSOLUTE MAXIMUM DEFORMATION OF SYSTEMS
SUBJECTED TO A HALF-CYCLE ACCELERATION PULSE
Uniasped. Elastic Systems ; Continuous Input Acceleration Functions
6-12
POUR-WAX LOG,
(VI
UO!|DJ9|900V punOJQ UinUJJXDft %
890ft 10 U0!|DJ9|990V UJnuijXDft " V
6-14
2.14a COMPARISON CP SYNTHESIZED AMD ACTUAL SPECTRA CORRESPONDIWJ TO A PULL-CYCLE ACCELERATION PULSE
Undaaped Elastic Systems; Acceleration Function Composed of Two Half -Sine Pulses
B
UO!4DJ8|800V punOJQ UinUJjXDft %
880^ IQ uoi|DJ8|893v uinuiixofl * V
6-15
2 .l4b COMPARISON CP SYNTHESIZED AND ACTUAL SPECTRA CORRESPONDING TO A FULL-CYCLE
ACCELERATION PULSE — Undamped Elastic Systems; Sinusoidal Pulse
Value of
PIO. 2.15 DEFORMATION SPECTRA FOR UNDAMPED ELASTIC SYS1B4S
SUBJECTED TO A VERSED-SINE VELOCITY PULSE
6-1 6
°A U0!|DJ9|900\/ punojQ umwixD^ °A
7^3 * 88Dfl JO UOIjDJ9|900V * T
6-17
2.16 SPECTRA FOR MAXIMUM AND MINIMUM ACCELERATIONS OF THE MASS
Undanped Elastic Systems Subjected to a Versed-Sine Velocity Pulse
6-18
6-19
FIG. 2.17b SPECTRA FOR ABSOLUTE MAXIMUM AND ABSOLUTE MINIMUM DEFORMATIONS
Undamped Elastic Systems Subjected to a Skewed Versed-Sine Velocity Pulse
AjpO|9A-opn»Sd A
®
O
>
6-20
FIG. 2.18 DEFORMATION SPECTRA FOR UNDAMPED ELASTIC SYSTEMS SUBJECTED TO SKEWED VERSED-SINE VELOCITY PULSES
t
FIG. 2.19 SPECTRA FOR MAXIMUM AND MINIMUM ACCELERATIONS OF THE MASS
Undamped Elastic Systems Subjected to a Parabolic Velocity Pulse
Aj!00|9/\ punojQ uinuMXDfl °*
A|iOO|0A-oPn®sd * A
6-22
2.20
AjpoidA punojg uunuuixo^ °A
A|!00|»A-opn98d 8 "A
B
6-23
2.21 EEFOFMATION SPECTRUM FOR UNDAMPED ELASTIC SYSTEMS SUBJECTED TO A SKEWED SIHJSOIDAL VELOCITY PULSE
Aj!90|9a punojo ujnuijxow °A
A|!00|9A-opni«d * 7\
6-24.
A4!OO|0/\ punojQ LununxDft °A
X4!OO|9A-opn0S^ 8 A
6-25
6-26
PIG. 2.22c EEFOFMATION SPECTRA FOR UNDAMPED ELASTIC SYSTEMS SUBJECTED TO TRIANGULAR VELOCITY PULSES
(9|oos 6o-|) /\ ‘^ijooieA-opnescj
6-27
2.23 DESIGN SPECTRUM FOR THE ABSOLUTE MAXIMUM DEFORMATION OF SYSTEMS SUBJECTED TO A HALF-CYCLE
VELOCITY PULSE — Undamped Elastic Systems; Continuous Input Acceleration Functions
6-28
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mam
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U!Ulnd
*ou*nd
*o »n|DA
FIG. 2.25b SPECTRA FOR ABSOLUTE MAXIMUM AND ABSOLUTE MINIMUM DEFORMATIONS
Undamped Elastic Systems Subjected to a Skewed Versed-Sine Displacement Pulse
CO
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K ft
B
3 g
VO
CM
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A4!00|9A-opnesd * A
6-31
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6-52
PIG. 2.27
(8|00s 6<n) a ‘A*!00|9A-opnesd
6-33
Undamped Elastic Systems; Continuous Input Acceleration Functions
A4j30|9/\ punojQ lunuujXD^ °A
A4po|9A -opnesd ' X
6-35
200 SPECTRA FOR MAXIMUM AND MINIMUM DEFORMATIONS —Undamped Elastic Systems Subjected
to a Full -Cycle Displacement Pulse; Velocity Diagram consists of Three Half -Sine Waves as Shown
islinll
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A|ioo|9A-opn0Sd A
6-36
A|!00|9a punojc) ujnujjXD^
A|ioo|9A-opn0Sd * X
.52
(•|00S Bon) A *^!00|®A-opnt«d
6-59
2.35 DESIGN SPECTRUM FOR THE ABSOLUTE MAXIMUM DEFORMATION OF SYSTEMS
SUBJECTED TO A FULL-CYCLE DISPLACEMENT PULSE
Undamped Elastic Systems; Continuous Input Acceleration Pulses
6-40
PIG. 2.54 DEFORMATION
o
A*!00|9a punojQ ujnuMXDw °A
A4!00|9A-opne«d * ”a
/^jOO|»A punojp ainuijXD^j ^ ^
Al!00|8A-°Pn98d A
6-42
FIG. 2.36 COMPARISON OF SPECTRA FOR ABSOLUTE MAXIMUM AND ABSOLUTE MINIMUM DEFORMATIONS
Damped Elastic Systems Subjected to a Versed-Sine Velocity Pulse
Damping
6-44
6-45
6-46
4.0
4.0
l i i l l
A|j90|9A punojQ wnumxDyy
A|;90|9a -opnesd
Damping Factor,
A4|30|9A punojQ uinuijxo^ °A
”” 44|00|9a -opnas^ ’ "a
6-50
FIG, 2.U2 DEFORMATION SPECTRA FOR DAMPED ELASTIC SYSTEMS SUBJECTED TO A FULL-CYCLE DISPL
Velocity Diagram .Consists of Three Half-Sine' Waves as Shown
6-51
Volue of t,f
FOR DAMPED ELASTIC SYSTEMS SUBJECTED TO A FULL -CYCLE SINUSOIDAL DISP1
4.0
o •
o d w
A|ioo|9a punojg uinuijxo^
A|!00|9A-opn9Sd
6-52
PIG. 2.44a DEFORMATION SPECTRA FOR DAMPED ELASTIC SYSTEMS SUBJECTED
TO A HALF-CYCLE PARABOLIC VELOCITY PULSE
6-53
Pseudo-Velocity
Maximum Ground Velocity
FIG. DEFORMATION SPECTRA FOR DAMPED ELASTIC SYSTEMS SUBJECTED
TO A FULL -CYCLE PARABOLIC VELOCITY PULSE
6-^4
Pseudo -Velocity
Maximum Ground Velocity
8.0
Value of t,f
FIG. 2.44c DEFORMATION SPECTRA FOR DAMPED ELASTIC SYSTEMS SUBJECTED
TO A SEQUENCE OF THREE PARABOLIC VELOCITY HALF -CYCLES
6-55
FIG. 2.kkd DEFORMATION SPECTRA FOR DAMPED ELASTIC SYSTEMS SUBJECTED
TO A SEQUENCE OF FOUR PARABOLIC VELOCITY HALF -CYCLES
6-56
For Secondary Pulse
Assuming that t2/td= 1/2 '
9S|nd AjDajUd jo Ajioo|9a wrmjjxow 'a
______________ ~
6-57
9S|nd Ajoujud J° ^i!°°laA _ _a
A4!DO|9A-oPn9Sd A
9S|nd AjdujUcJ jo A|!00|9a wnuiiXD^ 'a
A4!00|9A-opn9Sj “ "/\
6-59
0.178 g
6-60
6-61
FIG. 2.48 EL CENTRO, CALIFORNIA EARTHQUAKE OF MAY l8, 1940, N-S COMPONENT
ng Factor,
Ajpo|8A punojg uinuijxo^ °A
A4!00|9A-opne8d 8 *A
6-62
500
6-63
304 007 Ql 0.2 03 05 07 I 2 3 5 7 14
Undamped Natural Frequency, f, cps
FID. 2.50 SPECTRA FOR TIME OF MAXIMUM DEFORMATION FOR ELASTIC SYSTEMS SUBJECTED TO THE
-Damoin
A|»ooi9/\ punojp ujnmfXD^ °A
AjpojeA-opnasd s A
6-64
OOO ONtf) rO cm
^ fO CM -
998 **J (U0!|OUJJO^90 UlflUJIXOfl JO 9UJIX
8
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6-65
AjpoiaA punojp mnuiixow _
A*!00|»a - opnasd * A
6-68
Damping
6-69
Natural Frequency (Log Scale)
FIG. 2.57 RECOMMENDED FORM FOR PLOTTING SPECTRA FOR VARIOUS RESPONSE QUANTITIES
(Continued on Next Page)
6-70
Natural Frequency (Log Scale)
FIG. 2.57 (Continued)
6-71
6-72
6-73
°A
j- jo 9n|0A
6-75
6-76
X|jDO|«A »AUO|ay
vjpojaA-opnetd
8
Cm
6-77
6-78
6-79
88Dft jo uo!jdj8|800v ejnjosqv X
UO!JDJ8|8DOV“Opn88d S “v
6-80
2.62b COMPARISON OP SPECTRA FOR ABSOLUTE ACCELERATION OF MASS AND PSEUDO-ACCELERATION
Elastic Systems Subjected to a Versed-Sine Velocity Pulse
6-81
6-82
2.0
6 1 1 1 1 -7T
- 6
CVi
o
X
jo en|DA
6-83
o
..V::
fO
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0
d
d
UT-1
200
Damping Factor, /3=0
Pulse l
Pulse 2
(a) Half - Cycle Velocity Pulses
Full -Cycle Velocity Pulse
t
y
a
A
A-
y
1 » t
? A
\
r
♦. I/.
\
AJ
Pulse 4
r ”1 r
Pulse 5
(c) Multiple- Cycle Velocity Pulses
fio. 3.2 smpsa pulses cobukhbd
FIG. 30 COMPARISON OF MAXIMUM DEFORMATIONS OF ELASTIC SYSTEMS AND
ELASTOPLASTIC SYSTEMS WITH Uy « O.25 Uq —Undamped System*
Subjected to a Half -Cycle Parabolic Velocity Pulse
Value of V/y during Forced and Free Vibration
no.
.5 SPECTRA FOR MAXIMUM MTOHMATIOB WRI® FORCED VEBRATIOR ABD FREE VIBRATIC*
Undbtfved Elastic SyataM subjected to * Half -Cycle Parabolic Velocity Pulse
6-91
Reduction Factor, c s
Parabolic Velocity Pulse
6-92
2
3 5 7
Ductility Factor, fi
21
3i
PIG. J .6b RELATION BETWEEN REDUCTION FACTOR AND DUCTILITY FACTOR
Undamped Elastqplastic Systems Subjected to a Multiple -Cycle
Parabolic Velocity Pulse
FIG. 3.7a COMPARISON OF MAXIMUM DEFORMATIONS OF ELASTIC SYSTEMS
AND ELASTQPLASTIC SYSTEMS WITH Uy ej,0*25.
2 Percent Critical Damping Subjected to the Eureka Quake
6-94
Value of
TIG. 3.7b COMPARISON OF MAXIMUM DEFORMATIONS OF ELASTIC SYSTEMS
AND QtASTQPLASTIC SYSTSB WITH Uy - 0.23 Systems with
2 Percent Critical Dealing Subjected to the El Centro Quake
6-95
6-97
FIG. 3.8 (Continued)
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C CMP ARISON OF MAXIMUM DEFORMATIONS OF ELASTOPLASTIC
AND BILINEAR SYSTEMS WITH ^ - < 3.25 ' £ b£Uftke
2 Percent Critical Damping Subjected to the Eureka quake
FKJ. J.IO EFFECT OF STIFFHESS PARAMETER Wfe. OH MAXIMUM
n^OTMATIOH OF BHIHEAR SYS»?TOI TWO PSRCSHT
CRITICAL DAMP HO SUBJECTED TO THE EUREKA QUAKE
6-99
0.03-
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6-101
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FIG. 3.12b EEFOiWATION SPECTRA FOR ELASTOPLASTIC SYSTEMS WITH TEN PERCENT
CRITICAL DAMPING SUBJECTED TO A HALF-CYCLE PARABOLIC VELOCITY PULSE
6-103
FIG. 3.13a DEFORMATION SPECTRA FOR UNDAMPED, ELASTOPLASTIC SYSTEMS
SUBJECTED TO A HALF-CYCLE VELOCITY PULSE
Al!30|SA punojp uinui|xp^ _
A4!OO|0A“OPn®9d * A
6-104
PIG. 3.13'b DEFORMATION SPECTRA FOR ELASTOPLASTIC SYSTEMS WITH TEN PERCENT
CRITICAL DAMPING SUBJECTED TO A HALF -CYCLE VELOCITY PULSE
A|;oo|9a punojp uinmiXD^ °A
A4!00|9A-opn98d * "A
6-105
FIG. 3.14a DEFORMATION SPECTRA FOR UNDAMPED, ELASTOPLASTIC SYSTEMS
SUBJECTED TO A FULL-CYCLE PARABOLIC VELOCITY PULSE
Ductility Factor , fi- 1 —
A|IOO|9A punojQ mntutxoft ^
A4!00|«A-opn©6d " A
6-106
FIG. 3.14b DEFOFMATION SPECTRA FOR ELASTOPLASTIC SYSTEMS WITH TEN PERCENT
CRITICAL DAMPING SUBJECTED TO A FULL-CYCLE PARABOLIC VELOCITY PULSE
(•loos 6o-|)
A ‘A*!90|»A-opn»8d
6-107
FIG. 3.15 APPROXIMATE DESIGN RULE FOR CONSTRUCTION OF DEFORMATION SPECTRA
FOR ELASTOPLASTIC SIS3TMS-- See Text for Limitations
A»pO|»A puno«9 mnuitxo^ °A
A4!OO|9A-opn0«d A
6-108
3.16 EEFOFMATION SPECTRA FOR ELASTOPLASTIC SYSTEMS
TO A PARABOLIC VELOCITY PULSE WITH THREE HALF
Ductility Factor,
6-109
FIG. 3.17 DEFORMATION. SPECTRA FOR ELASTOFLASTIC SYSTEMS SUBJ]
TO A PARABOLIC VELOCITY PULSE WITH TWO FULL-CYCLES
-Ductility Factor,
A|ioo|9A'-opn8Sd A
6-110
ty Factor, /i.=l
6-111
APPENDIX A
TABULATION. OF NUMERICAL SOLUTIONS
la this Appendix is tabulated a portion of the numerical data used
to construct the response spectra presented in the body of the report. These
data were evaluated on the TLLIAC using the iterative scheme of numerical
Integration described in Ref. 11, with the acceleration of the mass assumed
to vary linearly within each step of integration. A description of the
computer program used is available in Ref. 12. The data for the remaining
spectra were obtained analytically by a formal solution of the governing
differential equation of motion.
For the elastic systems, the response quantities evaluated were the
absolute displacement, absolute velocity and absolute acceleration of the mass,
and the relative values of the displacement, velocity and acceleration between
the mass and the ground. Both the maximum positive and the maximum negative
values of these quantities were evaluated together with their respective times
of occurrence. Some of the deformation spectra presented in this report were
determined from these results by application of the analogies described in
Section 2.4. For example, the deformation spectra in Figs. 2.?8 through 2.40
were determined from the values of u presented in Table A. lb by noting that
the values of u/xo this table are also equal to wne values of u/yQ for an
input motion for which the shape of the displacement diagram is identical to
that of the velocity diagram shown below the table heading. Similarly, the
spectra in Figs. 2.42 were determined from the values of u given in Table A.lc.
For the inelastic systems, only the absolute displacement of the mass and the
spring deformation were computed.
A-l
For the pulse-like excitations, the time interval of integration used
in these computations was l/50th of the undamped natural period of the system,
whereas for the earthquake motions it ranged between l/jOth and l/70th of the
natural period. In addition to these time intervals, the response of the system
was evaluated at the times for which the ground acceleration was an extremum.
It should be noted that the tabulated data may not be accurate to the number of
significant figures reported. One of the sources of inaccuracy is the fact
that the response of the system is evaluated at discrete times. Since the
maximum response may occur between the times at which the response is evaluated,
in the absence of any other inaccur: cles, the computed maxima must be
numerically smaller than the actual maxima. The detailed data follow.
A-2
Elastic System Subjected to a Sequence of Two
Half-Sine Ground Acceleration Pulses as Shovm
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TABU A Oa( Continued) VALDES OF ABSOLUTE
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A- II
TABLE A.lc MAXIMUM AND MINIMUM VALUES OF RELATIVE ACCELERATION, u
Elastic Systems Subjected to a Sequence of Two Half-Sine
Ground Acceleration Pulses as Shown
A- 12
Denotes extremum occurring during free vibration.
TADLE A.lc MAX I HUM AMD MINIMUM VALUES OF RELATIVE ACCELERATION, « (CONTINUED)
CONh
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A- 13
ACCELERATION, G (CONTINUED)
TABLE A.lc MAXIMUM AMO MINIMUM VALUES OF RELATIVE ACCELERATION, U (CONTINUED)
A- 15
1
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A- 16
TABLE A. Id MAXIHUM AND MINIMUM VALUES OF ABSOLUTE VELOCITY OF MASS,
Elastic Systems Subjected to a Sequence of Two Half-Sine
Ground Acceleration Pulses as Shown
•x
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co in —
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A- 17
Denotes extremum occurring during free vibration.
OF ABSOLUTE VELOCITY OF HASS, x (CONTINUED)
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s!
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A- 19
TABU A. Id MAXIMUM AND MINIMUM VALUES OF ABSOLUTE VELOCITY OF MASS, % (CONTINUED)
81
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TABLE A.le MAXIMUM AND MINIMUM VALUES OF ABSOLUTE ACCELERATION OF MASS,
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OOONt
CM Ol CM —
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kfi co oi
115
242
394
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e e e o o o
i i • i i i
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i i i i i
o © o o
• iii
T T T
1.268
1.227
1.189
1.1S1
1.117
1.083
1.170
1.166
1.126
1.071
1.099
0.0100*
0.0224*
0.0610*
0.223*
0.429*
* * *
= 283
%o f. cm
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1.005*
*.125*
1.245
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1.007
1.031
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888 88
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88888
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1.071
1.067
1.062
1.057
1.053
1.050
1.046
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1.078
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1.073
1.063
1.064
82
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1.121
1.201
1.264
1.352
1.400
1.419
1.419
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1.208
1.303
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1.536
1.509
1.475
1.437
1.399
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1.279
1.380
32883
t ION0 44
• • • • •
1.623
1.591
1.551
1.510
1.465
22888
t
• • 0 • •
1.213
1.176
1.168
© cm
N «2
in Co
22888
r- oo oo r>* ©
,466
223
953
670
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1.408*
1.481
1.570
1.689
1.749
1.767
1.758
1.732
1.693
1.649
1.600
1.549
1.500
1.450
1.402
1.356
1.311
1.268
1.227
1.189
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nionvoi
o-nn*
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A- 2ft
TABLE A. 2a MAXIMUM AND MINIMUM VALUES OF PSEUDO VELOCITY
a*
V
i_
ji
c
»-
$
M-
-C
O
i/>
V
(A
u
<T>
c
V
<A
3
cr
(A
V
t A
3
a.
<D
o
8
n
•W
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<0
41
u
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<u
O
—
41
41
U
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u
<3
<
-o
c
6
3
s
0
y
u
a
>-
CO
4)
C
—
00
w
1
IA
4-
<0
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U 1
X
§
l
®J
§
o'
i
o
«m]
o’
I
«DJ
Cj
l
X
2
a
£
g*
f
3
&
C
*i
3
a
>?
o
>
o
■>*
©
o
i
<QJ
C
’£
3
a
x
3
l
c
3
a
o
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o
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.>?
!.>°
»°
O'
«
CQI
s
l\
3
a
o
SON
o Co oo o
cm om 3 3
3
OOOOO}
CO CO — ©
1/)
m so o>
o> co — h*
© — om cm
on n«
o>
in oo —
CO CO CO CO
00
roonn
CM
CM
CM CM CM
do oo
dodo
o
dodo
o
o
odd
till
i i i i
1
i i i i
1
i
i i i
co r- CM CT>
00 CO 00 Q>
o
in oo ao cm
00
to
CM VO 00 00
im oo r» o
& r* 3 om
o>
in — oo
in
O O — CM
oo of * m
CO
<•> CO CM
CM
* • • •
• • • •
•
• • « •
•
• • •
•
o o o o
o o o o
o
o o o o
o
o o o
o
$
fs
3 3 — oo
t
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o
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e ovo
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in
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$
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3
o
CO 00 o
o
• —
CM
co
in
m
in
in
m
3 co co
CO
i •
•
•
» • • •
•
•
•
•
•
*
• • •
•
o
o
o
o o o o
o
o
O
o
o
o
o o o
o
1
1
1
till
1
1
1
•
1
1
1 1 •
1
i O' p
V vfi vfi in
CM 3 CM — —
in — vO in
in
i % ^
o 3 m rs
O' in o' cm in
00 1" 00 CM 00
O — CO
in si r*> oo
oo oo i" no
in co co
CM
• • •
• • • •
• • ♦ • •
4 t • •
•
o o o
o o o o
o o o o o
o o o o
o
co r*> in
* *
♦ * *
O sO sO 00
3 O' CM CM
3'C'ON-
00 O' 3 O' N
— 'So —
O NO M0 o
NO 00 3 vo CM
O 3 — 0M 00
O — 00 3
• • • «
in in vo o
i • t i
OM CO OO — O'
r" m in 3 oo
• It!
o o o o
till
1 • 4 •
© © © —
till
T i i T ?
d d d o o
i i i i i
3s 00 * *
in n mo
— 3 NO NO
O O — 00
00 3 O 3
— t" © o>
SO 00 — CO
476
424
311
175
036
in cm — in o
o CO — 00 O'
a> so in co cm
o o o o
o d — —
o o o o d
-0.1714
-0.323
-0.449
-0.548
-0.726
-0.982*
-1.485*
-1.867*
-2.032*
-1.937*
-1.607
-1.152
-0.713
-0.609
-0.650
-0.481
-0.428
3
O' * *
i uo o' im
1 3 C" O'
0—00
670*
985
279
688
vONn3-
NNMNO
CO si 3 CO
cm o og ag so
CO si 3 00 si
— oo si 3 co
o o o
o o — —
— dodo
00
oo — in
co vo
t — r» 3
* * * *
so © 00 ©
S O' CM 3
* * +
CM O — 00 00
— — oo 3 m
n ia n Oi n
so co O' in
O' in o cm in
p- so oo in 3
o o o
1 1 I
do — —
lit!
CMCMCM-*-
1 1 1 1 1
©dodo
i « i i i
t CD & in
1 ©2?
S N (s 3
— in co r~
Nona
* *
co m 3 oo 3
in M oo n op
O — O sO 3
O & S 3 CM
co #i Mn 3
ooo
CM CM CM — —
— dodo
vO oo
s*> M9 p 00
O O' 3 oo
— — r> 3
♦ * * *
CM — O CM
sO co co —
n — w n
* * *
co — cm O r"
o in oo <o O'
O' — O' 3 so
*
rs in co oo oo
so O' — — a
O f" — fM 3
dodo
i i i i
O — — CM
till
CM CO CM CM —
1 1 1 1 1
— o — o o
i i « i i
* * ^ *
— CM * C"
c> cm 3 in
— in o 3
O O CM 3
* * * *
OM — © CM
s£ CO CO —
n - m n
* * _
CO — CM SO 3
omoocM —
O' — O' 3 N-
*
to SO 3 in SO
O i/l N N N
in — o so m
dodo
d — — CM
OM CO CM CM —
—■— — do
in in in
— om in im
o o o o
in
o cm m o
— — — OM
m o in p m
CM co co 3 3
o o o © o
m si O' oo o>
dodo
dodo
d d d d o
o o o o o
I*
A- 2 7
Denotes extremum occurring during free vibration.
VALUES OF PSEUDO VELOCITY (CONTINUED)
205
191
179
169
159
»«vO# n
nNN
® 28S|
3 = 82
8888 !
-0.
-0.
-0.
-0.
-0.
??>???
o o o o o
i i i i T
dodo
iii'
IA« OlA n
co — o co
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35388
O N N N n
CM — — O O
00 — CO
is Si :
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o o o o d
o o o o o
o d o o
OWIANN
h^NOOl
NNNN-
178
166
155
146
137
O « N •“'O
MN--0
M N O (71
CM n CM CO in
oot Nvom
— o o o o
OOOOO
1 i 1 1 1
ooooo
1 1 1 1 1
o o o o d
i i i i i
o o o o o
i i i i i
— n ntf n
in co — o> co
NNN--
NO O M IA
CM CM — © O
NO NO NO NO
— to — cm in
OISMOlfl
wOOOO
ooooo
o o o o o
OOOOO
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C* f* f" — 00
tOMAN
n n n n n
® ow-g
o o> r* ^ *
oo oi o no
CO CM CM — O
eg o> oo o
Nan no
onMon
— o o o o
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11(11
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1 • 1 1 1
o o o o o
i i i i i
o o o o o
i i i i i
N0 in — CD CO
Man®*
NNNN —
nognn
r*. Co % co co
r*. — no — \o
CM CM — — O
oo oo r- f"
— co — cm »n
o CO M3 in
— o o o o
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|
5
i
1
i
a
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s
— o in in w
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CO CO CO CM CM
in in cm oo
cm o oo in
CM CM — — —
v0 in in no op
4 n N — o
in co cm oo
— op no in in
OQNain
— © o o o
d o o o’ o
i i i i i
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t i i i i
o o o o o
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CM 00 01 — o
co m in cm o»
CO CM CM CM —
173
164
155
144
.135
•5 n© o n cm
A CM CM — O
w 8S
ottMOin
— o o © o
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N CO CO O -
co n n na
« CO CO N CM
— - O ON O CM
in # CM CM —
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— o o o in
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1 1 1 1 1
o • o o o
1 O l I I
o o o o o
i i i i i
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1 1 1 1 1
r* r* cm go cm
8 J22*
co n N CM ^
cM-maa
r* »m m5 in #
NWNfNt
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- ? N m3
28588
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ooooo
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ao cm co — r»
— 0) * —oo
>0 O' CO CO CM
250
232
211
188
196
8*382
co mt in
OO CM CO 00 P
OOOOO
i i T i i
o b b o o
i i i t •
o o o o o
i i i i i
o o o o o
i i i i i
«
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co co in n —
m ♦ CO CM CM
m m co >o ©
0)0 0) 00 00
at oi na
mJ *n-
o m 3 ? 8
= 2588
ooooo
d o o o o
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o d d o o
00
10
20
30
40
50
60
70
80
90
8283?
88888
CM CM CM CM CM
CM CO CO * *
A- 28
TABLE A. 2b MAXIMUM AMD MINIMUM VALUES OF RELATIVE VELOCITY,
Elastic Systems Subjected to a Sequence of Three
Half-Sine Ground Acceleration Pulses as Shown
-0.0547
-0.0861
-0.150
-0.199
-0.236
-0.286
-0.356
-0.461
CM O r» Q M>
cm m in in co
in in in in m
.....
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1 1 1 1 1
-0.519
-0.479
-0.441
-0.407
-0.377
0.0634
0. 1063
0.204
0.283
0.343
0.386
0.417
0.451
0.463
0.461
0.451
0.439
0.424
0.409
0.380
0.354
0.334
0.317
• •
-0.0924
-0.171
-0.236
-0.292
-0.337
-0.460
-0.759
in ^ 00 A XT
in CM A — CM
A O A A 00
• • • • •
0 — 000
1 1 1 1 I
A CM CM A in
co cm in a in
r*. m in *
.....
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i i i i i
fx
i mo in
• o cm <e
— CM CO
455
549
627
732
785
801
796
778
755
730
680
634
591
548
o' o o' oooo oeooe ooooo
in a oo vo
O © — (M
• • • •
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noMfl
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n o oo oo vo n
A — © &> — COh-NOt
vb A 00 — QlO MO I/)
• •••••
— ■ - OOOOO
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vooneo
o-n n
A O 00
* — in
m r*. oo
* *
-••IfiNN
nON*-oi
N O < < N
ix. in < — a
vOOl-N-
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© o © —
00 — 00
i a A n*
i o — CM
• • •
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* * *
h O MA
in n a O
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• • • •
© © © —
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cm o oo in co
r><cMO —
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• . • • •
CM CM CM CM —
I I I I •
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noiOMO
• • • • •
— o — © o
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r- d cm no
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n v in
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AOOOQ'C —
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CM — CO A 00
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CM A O A CO
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CM CM CO CM CM
•0.0S99
-0.0996
-0.197
-0.290
-0.378
-0.566*
-0.918*
-1.849*
-2.903*
-3.780*
-4.175*
-3.881*
-2.874*
-2.271
-1.907*
-3.007*
-1.744
-1.519
0.0610
0.1048
0.236
0.409
0.624
0.863
1.104
1.849*
SEES 5
e e • e •
ON
2.095
2.009
3.007*
1.730
1.545
m in m
— cm in r*«
oooo
in
o cm in o
— — — CM
in o m o in
wnnee
Q O O O O
in <2 IX 00 A
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A- 29
Denotes extremum occurring during free vibration.
I
Ml
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s
at
*
a
3
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i
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1
a
<£
3
2
— r- r<- q «*
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S$8«2
8 2 2 $ B
— £ 00 *
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o o o d
iiii
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«<««•»
2 2 S» 31 JS
N N - - -
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doddo
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OVOI522
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i i i i i
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vp in vo — p
8??88
bnoon
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22882
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8228?
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nnNNN
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doddo
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n N N N N
7????
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21552
moo «B«
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?2888
22822
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— dodo
doddo
o o d d o
o d o o o
BOh|@
»n«CN
51121
ftonft t
Jo Co ? 3 ?
B^NNOO
f» — r-oo —
nm Nrt n
77???
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s2si§
r* « & o w
S8383
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Jo 5 ? M 7
RSrSe
n m n n h
M-odd
o o o o o
o d o o o
o o O o o
8288? 22288 8228? 88882
A- 30
TABLE A. 2c SPECTRAL VALUES OF ABSOLUTE DISPLACEMENT OF THE MASS, X
Elastic System Subjected to a Sequence of Three
Half-Sine Ground Acceleration Pulses as Shown
y <
i . .
A
y i
,
y ,
^-^yo
A yo
/V
VT
H
IT’
V/
T’
t,
t. | t,
i t.
■ yo
, l( 1 1
♦d
fd
j" yo
X
V
Value of
P
fy (t> i t
r
o
3-0
8 - 0.05
8 - 0.10
0 - 0.20
0 - 0.50
0-1.00
0.015
1.00
m m
0.812*
• m
1.627
0.02S
1.00
0.922*
0.871*
0.814*
0.964
1.452
0.05
0.990*
0.910*
0.860*
0.802*
0.894
1.138
0.075
0.957*
0.885*
0.837*
0.781*
0.815
0.929
0.10
0.918*
0.852*
0.805*
0.750
0.736
0.779
0.125
0.872*
0.810
0.765
0.710
0.664
0.667
0.15
0.818*
0.761
0.718
0.662
0.599
0.580
0.20
0.700
0.652
0.615
0.562
0.490
0.455
0.25
0.586
0.547
0.517
0.472
0.407
0.371
0.30
0.487
0.457
0.432
0.397
0.342
0.311
0.35
0.405
0.382
0.363
0.335
0.292
0.267
0.40
0.339
0.321
0.307
0.286
0.252
0.232
0.45
0.286
0.273
0.262
0.246
0.221
0.206
0.50
0.243
0.233
0.225
0.213
0.195
0.184
0.60
0.180
0.175
0.171
0.166
0.158
0.152
0.70
0.138
0.136
0.135
0.134
0.132
0.129
0.80
0.1)0
0.110
0.111
0.113
0.113
0.111
0.90
0.0975
0.0974
0.0979
0.0988
0.0994
0.0986
1.00
0.0903
0.0892
0.0888
0.0887
0.0887
0.0883
1.10
0.0822
0.0813
0.0808
0.0804
0.0802
0.0799
1.20
0.0744
0.0739
0.0737
0.0734
0.0732
0.0730
1.30
0.0675
0.0675
0.0675
0.0675
0.0674
0.0672
1.40
0.0621
0.0623
0.0624
0.0624
0.0624
0.0623
1.50
0.0581
0.0581
0.0581
0.0581
0.0581
0.0580
1.60
0.0545
0.0544
0.0544
0.0544
0.0544
0.0543
1.70
0.0511
0.0511
0.0511
0.0511
0.0511
0.0510
1«80
0.0480
0.0481
0.0482
0.0482
0.0482
0.0482
1.90
0.045S
0.0455
0.0456
0.0456
0.0456
0.0456
Denotes extremum occurring during free vibration.
A-31
TABLE A. 2c SPECTRAL VALUES OF ABSOLUTE DISPLACEMENT OF THE MASS, X (CONTINUED)
Value of
o-o
0 - 0.05
0 - 0.10
0 - 0.20
0 - 0.50
8-1.00
2.00
0.0433
0.0433
0.0433
0.0433
0.0433
0.0433
2.10
0.0413
0.0412
0.0412
0.0412
0.0412
0.0412
2.20
0.0394
0.0393
0.0393
0.0393
0.0393
0.0393
2.30
0.0376
0.0376
0.0376
0.0376
0.0376
0.0375
2.40
0.0359
0.0360
0.0360
0.0360
0.0360
0.0360
2.50
0.0345
0.0345
0.0345
0.0345
0.0345
0.0345
3.00
0.0287
0.0287
0.0287
0.0287
0.0287
0.0287
3.50
0.0246
0.0246
0.0246
0.0246
0.0246
0.0246
4.00
0.0215
0.0215
0.0215
0.0215
0.0215
0.0215
4.50
0.0191
0.0191
0.0191
0.0191
0.0191
A-32
TABLE A. 2d SPECTRAL VALUES OP ABSOLUTE VELOCITY OF THE MASS, X (CONTINUED)
V _ Y,l“ of £
P ■ 0
P - 0.05
P - 0.50
P - 1.00
2.00
0.0747
0.0736
0.0730
0.0725
0.0722
0.0718
2.10
0.0705
0.0696
0.0692
0.0688
0.0686
0.0683
2.20
0.0667
0.0659
0.0656
0.0654
0.0653
0.0651
2.30
0.0626
0.0624
0.0624
0.0624
0.0623
0.0621
2.40
0.0591
0.0593
0.0595
0.0596
0.0596
0.0595
2.50
0.0571
0.0566
0.0569
0.0571
0.0571
0.0570
3.00
0.0480
0.0476
0.0474
0.0474
0.0473
0.0473
3.50
0.0404
0.0403
0.0404
0.0404
0.0404
0.0404
4.00
0.0356
0.0354
0.0353
0.0353
0.0353
0.0353
4.50
0.0313
0.0313
0.0313
0.0313
0.0313
0.0313
A- 34
MU A. 3* VALUES OP ABSOLUTE MAXIMUM DEFORMATIONS, u , AMD TEE ASSOCIATED TIMES
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A-38
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F-00 r-l IA IA MV
<0.
0 «
88333d
83R&3
3(8383
as,s«*a
t • a a a a
♦* 8
a a a a a a
H o 00 lACO f-
H H CM CM CM CM
* 3 *88
382?^^
Vi
%
88 CM w
MV t**
KVJ# ia VO 00
MV MV
J* (A MV
8
•
O
o o o o o o
o o o o d
H H CM CM A
-4 LA VO OO O g
*
•
g
MV
MV MV CM
t^«A^ MV
LA
LA UV CM (A
CMCMHHHO
• •••••
•
IAIAO t-MV.*
CM H H
MV CM CM H H
h d d d d
OOOOOO
a-J*6
TABUS A. 5 VALUES OF MAXIMUM AND MINIMUM PSEUDO-VELOCITIES AND ASSOCIATED TIMES
Blasto -Plastic Systems Subjected to Ground. Motion Shown:
P - 0 p - 0.10
V
t
JS2*
pu
max
t .
min
pumin
t
max
pumax
*min
Pumin
“s _
•
y9
_ *1
•
yo
h
s
yo
*1
*0
0.05
1
10,40
0.209
20.40
-0.209
10.00
0.155
1.00
-0.201
0.75
10.40
0.110
1.00
-0.206
9.6
0.070
1.00
-0.201
0.5
10.40
0.047
1.00
-0.207
0
0
1.00
-0.201
0.2
0
0
1.00
-0.208
0
0
1.00
-0.203
0.15
0
0
1.00
-0.208
0
0
1.00
-0.203
0,1
0
0
1.00
-0.209
0
0
1.00
-0.203
0.1
1
15.60
0,4l4
20.60
-0.4l4
5.20
0.308
1.00
-0,371
0.75
5.40
0.253
1.00
-0.395
5.00
0.151
1.00
-0.372
0,5
5.40
0.039
1.00
-0.398
0
0
1.00
-0.376
0.2
0
0
1.00
-0.405
0
0
1.00
-O.385
0.15
0
0
* 1.00
-0.410
0
0
1.00
-O.387
0.1
m
-
a* •
0
0
1.00
-O.389
0.2
1
3.00
0.804
5.50
-0.804
2.90
0.598
0.90
-0,615
0.75
3.00
O.663
0.90
-0.682
2.80
0.336
0.90
-0.622
0.5
3.00
0.242
0.90
-0.697
2.80
0.527
0.90
-0.641
0.25
0
0
0.90
-0.737
0
0
0.90
-0.676
0.15
0
0
1.00
-0.770
0
0
0.90
-0.695
0.1
0
0
1.00
-O.788
0
0
0.90
-0.708
0.15
m
•
-
0
0
1.00
-0.722
0.3
1
5.49
1.147
7.17
-1.147
2.08
0.854
0.80
-0.740
0.9
2.21
1.152
7.17
-0.913
2.08
0.858
0.80
-0.740
0.75
2.26
1.195
0.80
-0.840
2.08
0.698
0.80
-0.745
0,6
2.26
O.711
0.80
-0.846
2.08
0.469
0.80
-0.761
0.5
2.28
0.664
0.80
-0.860
2.14
0.304
0.80
-0.779
0.4
2.26
O.38O
0.87
-0.891
2.14
0.131
0.80
-0.804
0.3
2.35
0.061
0.87
-O.938
0
0
0.87
-0.844
0,25
0
0
0.87
-0.967
0
0
0.87
-0.867
0.15
0
0
0.94
-1.053
0
0
0.87
-0.924
0.1
0
0
0.94
-1.106
0
0
0.94
-0.961
0.05
0
0
1.00
D
-1.170
0
0
0.94
-1.005
0.4
1
1-75
1.424
3.00
-1.424
1.70
1.060
0.70
-O.790
0.9
1.75
1.431
3.00
-1.131
1.70
1.066
0.70
-0.790
0.75
1.80
1.483
0.70
-O.905
1.75
1.100
0.70
-O.790
0.6
1.90
1.485
0.70
-0.905
1.75
0.849
0.70
-0.801
0.5
1.90
1.199
0.75
-0.920
1.80
0.655
0.75
-0.822
0.4
1.95
O.856
0.75
-0.959
1.85
0.429
0.75
-O.863
TABLE A. 5 CONTINUED
8 -
0
&
- 0.10
V
i
tmax
^%ax
t .
min
^in
t
max
"W
t .
min
pumin
uo
4
yo
\
*0
‘i
yo
*1
yo
0.5
2.00
0.443
0.80
-1.032
1.90
O.176
0.80
-0.924
0.25
2.05
0.203
O.85
-I.O85
1.95
0.037
0.80
-O.967
0.15
0
0
0.90
-1.244
0
0
O.85
-1.075
0.1
0
0
0.90
-1.353
0
0
0.90
-1.144
0.05
0
0
0.95
-1.494
0
0
0.90
-1.231
0.5
1
3-52
1.617
4.52
-1.617
1.44
1.204
2.48
-0.879
0.9
1.52
1.283
2.52
-1.629
0.8
1.52
1.660
2.52
-O.927
0.75
1.56
1.689
0.64
-0.919
1.52
1.252
0.60
-0.794
0.7
1.56
1.724
0.64
-0.919
0.6
1.64
1.837
0.64
-0.919
0.5
1.68
1.696
0.64
-0.924
1.60
0.981
0.64
-O.818
0.4
1.72
1.342
0.68
-0.961
0.3
1.80
0.709
O.72
-I.050
0.25
1.84
0.596
O.76
-1.125
1.76
0.274
0.76
-1.002
0.2
1.80
0.253
0.80
-1.227
0.15
0
0
0.84
-1.362
0
0
0.80
-1.168
0.1
0
0
0.88
-1.540
0
0
0.84
-1.278
0.05
0
0
O.96
-1.773
0.7
1
4.08
1.736
3.36
-1.736
1.20
1.296
1.92
-0.946
0.9
2.64
1.744
1.92
-1.383
1.20
1.303
O.50
-0.750
0.75
1.24
1.809
O.52
-0.872
1.24
1.344
O.50
-0.750
0.6
4.16
1.967
O.52
-O.872
1.28
1.446
0.50
-0.750
0.5
1.36
2.165
O.52
-O.872
1.32
1.321
O.52
-O.759
0.4
4.26
1.887
O.56
-0.894
1.36
1.106
O.56
-0.802
0.3
1.52
1.514
0.60
-0.996
1.44
0.826
0.60
-0.905
0.25
3.00
1.247
0.64
-1.098
1.48
0.644
0.64
-0.990
0.15
3.16
0.365
O.76
-1.478
1.64
O.168
O.72
-1.254
0.1
0
0
0.84
-1.805
0
0
0.80
-1.448
0.05
0
0
0.92
-2.271
0
0
0.84
-1.699
1
1
1.00
1.274
1.50
-1.274
1.02
0.966
1.52
-O.705
0.9
1.00
1.286
1.50
-I.007
0.8
1.02
1.333
0.40
-O.761
0.75
1.04
1.373
0.40
-O.761
1.04
1.030
0.40
-0.653
0.6
1.08
1.591
0.40
-0.761
0.5
1.14
1.496
0.42
-0.777
1.14
0.962
0.44
-0.693
A- 48
TABLE A. 5 CONTINUED
V
1.25
1.50
2
P = 0 p = 0.10
i
uo
t
max
*1
pu
max
yo
Y
pumin
t
max
*1
pu
max
~*o
^min
*1
pumin
y0
0.4
1.22
1.450
0.46
-O.85O
0.3
1.54
1.358
0.54
-1.042
0.25
1.44
1.218
0.60
-1.220
1.36
0.607
0.60
-1.081
0.15
1.64
O.367
O.74
-1.873
0.1
0
0
0.82
-2.4i4
0
0
0.74
-1.759
1
0.80
0.815
0.34
-0.676
O.83
0.576
0.34
^0.580
0.9
0.82
0.823
0.34
-0.676
0.86
0.499
0.34
-O.585
0.75
0.86
0.746
0.34
-0.680
1.02
0.498
0.37
-O.618
0.6
1.04
0.756
0.34
-0.733
1.09
O.572
0.42
-O.707
0.5
1.12
0.410
0.38
-0.828
1.15
0.605
0.46
-0.820
0.4
1.23
1.066
0.43
-1.028
1.23
0.576
0.53
-1.000
0.3
1.38
1.039
O.50
-1 . 4i4
1.33
-.440
0.59
-1.241
0.25
1.47
0.843
0.59
-1.714
1.39
O.318
0.62
-1.446
O.15
0
0
0.66
-2.635
0
0
O.70
-1.895
0.1
0
0
0.78
-3.305
0
0
0.75
-2.176
0.05
0
0
0.93
-4.155
0
0
0.78
-2.499
1
0.66
O.566
0.29
-O.603
0.69
0.366
0.29
-O.518
0.9
0.68
0.457
O.30
-0.608
0.71
0.290
0.29
-O.523
0.75
O.78
. 0.293
O.32
-0.647
O.76
0.176
O.32
-0.556
0.6
1.01
0.191
0.37
-0.764
1.03
0.145
0.37
-0.648
0.5
1.10
0.532
0.43
-0.934
1.09
0.298
0.42
-O.772
0.4
1.24
0.831
0.52
-1.251
1.16
0.400
0.49
-0.973
0.3
2.06
0.841
0.62
-1.803
1.26
0.374
O.56
-1.284
O.25
1.49
0.556
O.67
-2.204
1.32
0.295
0.60
-1.489
O.15
0
0
0.80
-3.351
0
0
0.68
-2.013
0.1
0
0
0.86
-4.139
0
0
O.72
-2.341
0.05
0
0
0.94
-5.103
0
0
0.77
-2.716
1
1.00
0.637
1.25
-0.637
1.01
0.404
0.22
-0.424
0.9
1.00
0.643
1.25
-0.504
1.01
0.345
0.23
-0.428
0.6
1.01
0.668
0.23
-0.493
1.02
0.277
0.24
-0.444
0.75
1.02
0.651
0.23
-0.493
1.02
0.237
0.25
-O.459
0.6
1.04
0.464
0.25
-0.525
1.04
0.061
0.31
-0.690
0.5
1.06
0.266
0.29
-0.601
0
0
O.36
-0.690
0,4
0
0
O.36
-0.799
0
0
0.43
-O.925
0.3
1.12
0.086
0.48
-1.290
1.18
0.140
0.51
-1-297
O.25
1.24
0.475
0.55
-1.740
1.23
0.164
O.56
-1.546
0.15
0
0
0.72
-3.310
1.38
0.002
0.64
-2.190
0.1
0
0
0.81
-4.559
0
0
0.68
-2.590
TABLEAU CONTINUED
P - 0 p - 0.10
1
uo
' 1
yo
t .
min
*1
pumin
yo
t
max
*1
pu
max
yo
t .
min
*1
Pumin
yo
1
0.80
0.407
0.18
-0.4l6
0.82
0.227
0.18
-0.357
0.9
0.81
0.327
0.19
-0.419
0.84
0.180
0.18
-O.36I
0.75
0.84
0.196
0.21
-0.452
0.89
0.100
0.21
-0.390
0.6
1.01
0.096
0.26
-0.565
1.02
0.019
0.26
-0.483
0.5
1.06
0.069
0 33
-0.764
0
0
0.52
-O.625
0.4
0
0
0.42
-1.196
0
0
0.40
-O.883
0.5
1.22
0.082
0.55
-2.076
0
0
0.46
-1.302
0.25
1.3^
0.277
0.62
-2.768
0
0
0.52
-1.586
0.15
0
0
0.77
-4.858
0
0
0.61
-2.316
0.1
0
0
0.84
-6.345
0
0
0.66
-2.769
0.05
0
0
0.92
-8.193
0
0
0.70
-3.284
1
1.00
0.425
1.50
-0.425
1.00
0.241
0.16
-O.309
0.9
1.00
0.429
0.16
-0.358
1.01
0.202
0.16
-O.312
0.75
1.01
0.358
0.16
-0.363
1.01
0.119
0.18
-0.339
0.6
1.02
0.187
0.19
-0.409
0
0
0.23
-0.429
0.5
0
0
0.23
-0.511
0
0
0.29
-0.575
0.4
0
0
0.32
-0.793
0
0
0.37
-0.849
0.3
0
0
0.46
-1.550
0
0
0.46
-1.304
0.25
0
0
0.53
-2.261
0
0
0.50
-l.6l4
0.15
0
0
0.71
-4.747
0
0
0.59
-2.409
0.1
0
0
0.80
-6.704
0
0
0.63
-2.901
0.05
0
0
0.90
-9.266
0
0
O.67
-3.460
1
1.00
0.318
1.13
-0.318
1.01
0.170
0.12
-<0.242
0.9
1.00
0.321
0.12
-O.28O
1.01
0.140
0.12
-0.244
0.75
1.01
0.236
0.13
-0.288
1.01
0.077
0.14
-0.268
0.6
1.02
0.080
0.16
-0.338
0
0
0.19
-0.351
0.5
0
0
0.20
-0.452
0
0
0.25
-0.500
0.4
0
0
0.30
-0.802
0
0
0.33
-0.797
0.3
0
0
0.44
-1.813
0
0
0.42
-1.307
0.25
0
0
0.53
-2.783
0
0
0.46
-1.652
0.15
0
0
0.71
-6.185
0
0
0.55
-2.539
0.1
0
0
0.81
-8.847
0
0
0.59
-3.083
0.05
0
0
0.90
-12.32
0
0
0.64
-3.696
1
1.00
0.255
1.10
-0.255
1.00
0.131
0.10
-0.198
0.75
1.00
0.172
0.10
-0.239
1.00
0.056
0.12
-0.221
0.5
0
0
0.18
-0.411
0
0
0.22
-0.446
0.4
0
0
0.28
-0.818
0
0
0.30
-0.759
0.25
0
0
0.52
-3.302
0
0
0.44
-1.675
0.15
0
0
0.71
-7.620
0
0
0.52
-2.618
TABLE A. 5 CONTINUED
V
i
uo
0 =
0
0
= 0.10
t
max
*1
pumax
yo
^min
tl
pumin
. *' ’ "
y0
^max
tl
pu
max
•
yo
^min
*1
pumin
yo
7
1
1.00
0.182
1.22
-0.182
1.00
0.091
1.07
-0.146
0.9
1.29
0.172
0.07
-0.169
1.00
0.075
0.07
-0.148
0.75
1.29
0.108
0.08
-0.179
1.00
0.037
0.08
-O.163
0.6
0
0
0.10
-0.227
0
0
0.13
-0.230
0.5
0
0
0.15
-0.355
0
0
0.19
-0.375
0.4
0
0
0.26
-0.866
0
0
0.27
-O.709
0.3
0
0
0.42
-2.606
0
0
0.36
-1.298
0.25
0
0
0.52
-4.346
0
0
0.40
-1.696
0.3 5
0
0
0.71
-10.49
0
0
0.49
-2.705
0.1
0
0
0.80
-15.28
0
0
0.53
-3.315
0,05
0
0
0.90
-21.48
0
0
0.57
-3.998
10
1
1.00
0.127
1.05
-0.127
1.00
0.063
0.05
-0.104
0.75
1.00
0.067
0.06
-0.130
1.00
0.024
0.06
-0.118
0.5
0
0
0.13
-0.303
0
0
0.16
-0.313
0.4
0
0
0.25
-0.948
0
0
0.24
-O.663
O.25
0
0
0.51
-5.920
0
0
0.37
-1.706
0.15
0
0
0.70
-14.80
0
0
0.46
-2.757
TABLE A.6 VALUES OF MAXIMUM AND MINIMUM PSEUDO-VELOCITIES AND ASSOCIATED TIMES
o.i i 15.60 0.311 20.60 -0.311 5.20 0.232 0.80 -0.281
0.9 5.60 0.279 1.00 -0.297 5.20 0.185 0.80 -0.281
0.75 5-40 0.185 1.00 -0.298 5.00 0.114 0.80 -0.282
0.6 5.U0 0.089 1.00 -0.299 5-00 0.01+2 0.80 -0.282
0.5 5. to 0.021+ 1.00 -0.301 0 0 0.80 -0.283
0.4 0 0 1.00 -0.302 0 0 1.00 -0.284
0.3 0 0 1.00 -0.304 0 0 1.00 -0.286
0.25 0 0 1.00 -0.306 0 0 1.00 -0.287
0.15 0 0 1.00 -0.309 0 0 1.00 -0.290
0.1 0 0 1.00 -0.310 0 0 1.00 -0.291
0.05 0 0 1.00 -0.312 0 0 1.00 -0.295
0.2 1 3.00 0.612 5*50 -0.612 2.90 0.455 0.80 -0.502
0.9 3.00 0.616 0.80 -0.543 2.80 0.372 0.80 -0.502
0.75 3.00 0.452 0.8b -0.544 2.80 0.244 0.80 -0.504
0.6 3.00 0.265 0.80 -0.547 2.70 0.112 0.80 -0.508
0.5 3.00 0.138 0.80 -0.550 2.70 0.022 0.80 -0.512
0.4 0 0 0.90 -0.559 0 0 0.80 -0.517
0.3 0 0 0.90 -0.569 0 0 0.80 -0.524
0.25 0 0 0.90 -0.577 0 0 0.80 -0.528
0.15 0 0 0.90 -0.591 0 0 0.90 -0.559
0,1 0 0 0.90 -0.601 0 0 0.90 -0.546
0.05 0 0 0.90 -0.611 0 0 0.90 -0.555
0.4 1 1.75 1.156 5.00 -1.136 1.70 0.845 0.70 -0.750
0.9 1.75 1.142 5-00 -0.905 1.70 0.850 0.70 -0.750
0.75 1.80 1.185 0.70 -0.821 1.70 0.701 0.70 -0.752
0.6 1.85 0.950 0.70 -0.826 1.70 0.480 0.70 -0.745
0.5 1.85 0.681 0.75 -0.842 1.75 0.325 0.75 -0.759
0.4 1.85 0.404 0.75 -0.871 1.75 0.149 0.75 -0.787
0.5 1.90 0.104 0.75 -0.911 0 0 0.75 -0.822
0.25 0 0 0.80 -0.942 0 0 0.75 -0.844
0.15 0 0 0.80 -1.020 0 0 0.80 -0.900
0.1 0 0 0.85 -1.078 0 0 0.80 -0.956
0.05 0 0 0.90 -1.149 0 0 0.80 -0.978
TABLE A. 6 CONTINUED
= 0
e
= 0.10
V
u
JL
t
max
pu
max
"'"'min
pUmin
t
max
pu
* max
^min
pumin
u
o
*1
*1
\
*1
K
0.6
1
1.34
1.500
2.18
-1.500
1.31
1.117
2.14
-0.815
0.9
1.34
1.509
2.18
-1.191
1.31
1.123
0.60
-0.779
0.75
1.38
1.563
0.64
-O.896
1.34
1.160
0.60
-0.779
0.6
1.44
1.703
0.64
-O.896
1.37
1.015
0.64
-0.784
0.5
1.48
1.432
0.64
-0 . 906
1.37
0.811
0.64
-0.806
0.4
1.51
1.098
0.67
-0.941
1.41
0.584
0.67
-0.845
0.3
1.54
0.673
O.70
-1.019
1.47
O.3I8
0.67
-0.913
0.25
1.57
0.414
0.70
-1.080
1.51
0.164
0.70
-0.962
0.15
0
0
0.75
-1.264
0
0
0.74
-I.090
0.1
0
0
0.80
-1.399
0
0
0.75
-1.178
0.05
0
0
0.84
-1.587
0
0
0.77
-1.286
0.8
1
1.13
1.663
1.75
-1.663
1.10
1.239
1.73
-O.903
0.9
1.13
1.672
1.75
-1.321
1.10
1.244
0.55
-0.753
0.75
1.15
1.733
0.55
-0.871
1.13
1.286
0.55
-0.753
0.6
1.20
1.887
0.55
-0.871
1.18
1.362
0.55
-0.753
0.5
1.25
1.963
0.55
-0.872
1.20
1.173
0.55
-0.753
0.4
1.30
1.649
0.58
-0.902
1.23
0.941
0.58
-0.807
0.3
1.35
1.234
0.63
-0.991
1.30
0.651
0.63
-0.896
0.25
1.40
0-954
0.65
-1.075
1.53
OA78
0.65
-0.965
0.15
1.50
0.155
O.70
-1.366
1.45
0.051
0.70
-1.171
0.1
0
0
0.75
-1.609
0
0
0.73
-1.322
0.05
0
0
0.83
-1.946
0
0
0.75
-1.513
1
1
1.00
1.621
1.50
-1.621
1.00
1.209
1.50
-0.882
0.9
1.00
1.629
1.50
-1.288
1.00
1.216
0.50
-0.698
0.75
1.02
1.691
0.50
-0.810
1.02
1.260
0.50
-0.698
0.6
1.06
1.852
0.50
-0.810
1.06
1.363
0.50
-0.698
0.5
1.12
2.060
0.50
-0.810
1.08
1.260
0.50
-O.707
0
0.4
1.16
1.817
0.52
-0.832
1.12
1.074
0.52
-0.750
0.3
1.24
1.507
0.56
-0.932
1.18'
0.826
0.56
-0.855
0.25
1.28
1.275
0.60
-1.035
1.22
0.658
0.60
-0.943
0.15
1.42
0.448
0.68
-IA30
1.34
0.204
0.66
-1.220
0.1
0
.0
0.72
-1.776
0
0
0.70
-1.427
0.05
0
0
0.80
-2.285
0
0
0.74
-2.339
1.25
1
0.90
1.348
1.3c
-1.325
0.90
1.015
1.31
-0.727
0.9
0.90
1-358
2.11
-1,048
0.90
1.021
0A5
-0.622
0.75
0.91
1.423
0A5
-0.720
0.93
1.071
0A5
-0.622
0.6
O.96
1.606
0.45
-0.720
O.96
1.166
0A5
-0.622
0.5
1.01
1.703
0A5
-0.722
1.00
1.076
0.46
-0.640
0.4
1.06
1.607
0.48
-0.761
1.04
0.978
0.50
-0.701
A-53
TABLE A.6 CONTINUED
3
* 0
B
- 0.10
V
u
JL ■
t
max
^umax
"Sain
pumln
t
max
pumax
"Vin
pumin
uo
*1
f'l
^ " 1
*1
*0
tl
K
0.5
1.15
1.479
0.53
-0.903
1.10
0.815
0.54
-0.840
0.25
1.20
1.355
O.56
-1.044
1.15
0.685
0.58
-O.956
0.15
1.36
0.541
0.66
-1.594
1.28
0.259
0.64
-1.319
0.1
0
0
0.72
-2.078
0
0
0.67
-1.587
0.05
0
0
0.78
-2.777
0
0
0.72
-1.933
1.5
1
0.81
0.997
1.83
-0.871
0.82
0.762
0.4o
-0.549
0.9
0.82
1.006
1.18
-0.677
0.83
0.769
0.41
-0.549
0.75
0.84
1.075
0.41
-0.632
0.86
0.824
0.41
-0.549
0.6
0.90
1.195
0.41
-0.633
0.92
0.804
0.42
-O.563
0.5
0.96
1.211
0.43
-0.660
o.97
0.808
0.44
-0.611
0.4
1.03
1.307
0.47
-0.755
1.02
0.801
0.49
-0.715
0.3
1.14
1.340
0.53
-0.993
1.09
0.697
0.54
-0.919
0.25
1.20
1.214
0.58
-1.211
1.14
0.586
0.58
-1.071
0.15
1.37
0.311
0.67
-1.965
1.27
0.193
0.64
-1.512
0.1
0
0
0.73
-2.577
0
0
0.67
-1.816
0.05
0
0
0.79
-3-412
0
0
0.71
-2.190
2
1
0.65
0.480
0.35
-0.480
0.69
0.356
0.35
-0.424
0.9
0.67
0.408
0.35
-0.485
0.71
0.307
0.36
-0.428
0.75
0.79
0.362
0.38
-0.527
0.85
O.289
0.38
-0.465
0.6
0.95
0.712
0.44
-0.663
0.93
0.441
O.43
-0.573
0.5
1.04
1.050
0,49
-0.869
0.99
0.554
0.47
-0.715
0.4
1.14
1.215
0.55
-1.232
1.05
0.582
0.52
-0.940
0-3
1.26
0.939
0.62
-1.829
1.12
0.474
0.57
-1.268
0.25
1.32
0.540
0.66
-2.242
1.16
0.363
0.59
-1.475
0.15
0
0
0.73
-3.361
1.29
0.030
0.64
-1.994
0.1
0
0
0.77
-4.102
0
0
0.67
-2.312
0.05
0
0
0.83
-5.017
0
0
0.69
-2.674
2.5
1
0.88
0.358
O.31
-O.363
0.86
0.271
0.32
-0.327
0.9
0.88
0.275
0.32
-O.368
0.85
0.221
0.33
-0.332
0.75
0.62
0.095
0.35
-0.418
0.84
0.104
0.36
-0.376
0.6
0.90
0.214
0.42
-0.607
O.90
0.145
0.42
-O.517
0.5
1.02
0.819
0.49
-0.901
0.97
0.350
0.46
-O.705
0.4
1.13
1.183
0.55
-1.414
1.02
0.467
0.51
-0.987
O.J
1.26
0.921
0.62
-2.229
1.10
0.422
0.56
-1.384
0.25
1.32
0.457
0.66
-2.780
1.14
0.334
0.58
-1.630
0.15
0
0
0.74
-4.235
1.26
0.046
0.63
-2.224
0.1
0
0
0.78
-5.174
0
0
0.66
-2.577
0.05
0
0
0.83
-6.314
0
0
0.67
-2.975
A-5*
TABLE A. 6 CONTINUED
e
= 0 .
p
= 0.10
V
u
JL
t
max
pu
* max
"Sain
pumin
t
max
pu
meat
^min
pumin
uo
*1
"V
*1
*0
*1
*0
K
3
1
0.81
0.344
0.28
-0.273
0.82
0.266
0.29
-0.254
0.9
0.81
0.348
0.28
-0.273
0.82
0.247
0.30
-0.256
0.75
O.83
0.345
0.29
-0.275
0.85
0.165
0.34
-0.298
0.6
0.88
0.204
0.35
-0.352
0
0
o.4l
-0.467
0.5
0
0
0.42
-0.573
0.95
0.178
0.46
-0.692
0.4
I.03
0.822
0.50
-1.079
1.01
O.366
0.50
-1.026
0.3
1.51
1.196
0.59
-2.011
1.07
0.375
0.55
-1.480
0.25
1.59
0.888
O.63
-2.687
1.11
0.309
0.58
-1.754
0.15
0
0
0.72
-4.558
1.22
O.063
0.62
-2.4o6
0.1
0
0
O.76
-5.808
0
0
0.64
-2.787
0.05
0
0
0.82
-7.338
0
0
0.66
-3.209
4
1
0.75
0.159
0.25
-0.159
0.79
0.165
0.29
-0.164
0.9
0.87
0.290
0.33
-0.207
0.82
0.164
0.31
-0.174
0.75
0.95
0.256
0.43
-0.538
O.89
0.164
0.38
-O.305
0.6
1.04
0.737
0.51
-1.316
0
0
0.49
-0.630
0.5
1.14
1.284
0.57
-2.157
0.98
0.133
0.48
-0.960
0.4
1T 25
1.044
0.63
-3.311
1.03
0.278
0.52
-1.382
0.3
0
0
0.78
-4.824
1.08
0.271
O.56
-1.897
0.25
0
0
0.71
-5. 73*+
1.11
0.216
0.57
-2.190
0.2
0
0
0.77
-7.895
1.20
0.034
0.6l
-2.849
0.15
0
0
0.81
-9.184
0
0
0.62
-3.218
0.1
0
0
0.86
-10.68
0
0
0.64
-3.614
5
1
0.80
0.166
O.30
-O.134
0-79
0.139
0.29
-0.132
0.9
0.80
0.169
0.30
-0.134
0.80
0.134
0.30
-O.133
0.75
0.83
0.167
O.31
-0.136
0.85
0.107
0.35
-0.188
0.6
0.91
0.415
0.38
-0.262
0.92
0.151
0.42
-0.488
0.5
0.97
0.162
0.47
-0.896
0
0
0.46
-0.848
0.4
1.09
1.189
0.55
-2.108
1.00
0.134
0.50
-I.319
0.3
1.24
1.175
0.63
-4.007
1.04
0.226
0.53
-1.897
0.25
1.31
0.317
0.66
-5.252
1.07
0.207
0.55
-2.226
0.15
0
0
0.74
-8.417
1.14
0;074
0.58
-2.962
0.1
0
0
0.78
-IO.38
0
0
0.60
-3.371
0.05
0
0
0.84
-12.71
0
0
0.62
-3.810
7
1
0.78
0.115
0.27
-0.102
0.78
0.097
0.27
-0.097
0.9
0.78
0.118
0.26
-0.102
0.79
0.082
0.29
-0.105
0.75
0.81
0.057
O.31
-0.134
0.85
0.079
0.35
-0.214
0.6
0.91
0.090
0.41
-0.489
0.91
0.049
0.41
-0.529
0.5
0.99
0.670
0.47
-1.176
0.95
0.125
0.45
-0.900
0.4
1.23
1.286
0.55
>2.721
0.98
0.110
0.48
-1.423
A-55
TABLE A. 6 CONTINUED
p
« 0
p
- 0.10
V
u
JL
t
max
pu
* max
t .
min
pumin
t
max
pu
* max
^min
uo
*1
y0
h
*0
*i
yo
*1
0.3
1.24
1.419
0.63
-5.473
1.02
0.120
0.51
-2.058
0^25
1.30
2.320
0.67
-7.272
1.04
0.142
0.53
-2.413
0.15
0
0
0.74
-11.80
1.09
0.082
O.56
-3.196
0.10
0
0
0.78
-14.58
1.14
0.017
O.58
-3.625
0»05
0
0
0.84
-17.84
0
0
0.59
-4.079
10
1
0.73
0.070
0.27
-0.070
0.77
0.065
0.27
-0.066
0.9
1.56
0.075
O.23
-0.075
0.79
0.060
0.29
-0.076
0.75
1.37
0.232
0.36
-0.221
0.85
0.059
0.35
-0.219
0.6
1.46
0.069
0.46
-1.357
0.90
0.041
o.4l
-0.627
0.5
1.02
0.093
0.52
-2.966
0.94
0.027
0.44
-1.032
0,4
1.15
1.932
0.59
-5.419
0.97
0.073
0.47
-1.545
0.3
1.20
0.534
0.65
-9.312
0.99
o.oo4
0.49
-2.197
0.25
0
0
O.69
-11.87
1.01
0.056
0.51
-2.565
0.15
0
0
0.76
-18.10
1.05
0.069
0.54
-3.376
0.1
0
0
0.79
-21.81
1.08
0.033
0.55
-3.814
0.05
0
0
0.85
-26.09
0
0
0.56
-4.274
TABUS A. 7 VALUES OF MAXIMUM AND MINIMUM PSEUDO -VELOCITIES AND ASSOCIATED TIMES
Elas to -Plastic Systems Subjected to Ground Motion Shown:
y
h
... \
y
\ _
...
Xj U
. O
w| I
iVt
L 1 w. _ ±J
n n
P =
0
P
= 0.10
V
t
max
pu
max
"^min
^Umin
t
max
pu
max
^min
P\in
uo
*1
*0
*0
*1
*0
*1
K
0.05
1
6.00
O.O67
1.00
-0.206
5.20
O.O58
1.00
-0.201
0.75
24.00
0.029
1.00
-0.206
3.40
0.034
1.00
-0.201
0.5
2.80
0.015
1.00
-0.207
2.40
0.023
1.00
-0.201
0.25
2.40
0.006
1.00
-0.208
2.00
0.016
1.00
-0.202
0.15
22.40
0.003
1.00
-0.208
2.00
0.014
1.00
-0.203
0.1
22.40
0.002
1.00
-0.209
2.00
0.013
1.00
-0.203
0.1
1
•
0.255
1.00
-0.394
3.00
0.225
1.00
-0.371
0.75
2.80
0.160
1.00
-0.395
2.60
O.163
1.00
-0.372
0.5
2.60
0.097
1.00
-0.399
2.20
0.119
1.00
-0.376
0.25
2.40
0.044
1.00
-0.407
2.00
0.079
1.00
-0.383
0.15
2.20
0.026
1.00
-0.410
2.00
0.066
1.00
-0.387
0.1
2.20
0.018
1.00
-0.413
2.00
0.059
1.00
-0.389
0.2
1
7.30
0.943
9.80
-0.943
2.00
0.831
0.90
-0.615
0.75
2.40
0.972
0.90
-O.681
2.00
0.842
0.90
-0.615
0.5
2.30
0.725
0.90
-O.691
2.00
O.651
0.90
-0.626
0.25
2.30
O.388
0.90
-0.728
2.00
0.418
0.90
-0.662
0.15
2.20
0.236
1.00
-0.760
2.00
0.317
0.90
-O.685
0.1
2.20
0.159
1.00
-0.781
2.00
0.267
0.90
-0.699
0.3
1
8.51
1.855
6.83
-1.855
1.88
1.497
3.35
-1.190
0.9
1.94
1.809
3.48
-1.577
1.88
1.502
3.35
-b.924
0.75
1.94
1.839
3.55
-0.971
1.88
1.525
0.80
-0.740
0.6
2.00
1.899
0.80
-0.840
1.94
1.571
0.80
-0.740
0.5
2.00
1.971
0.80
-0.840
1.94
1.629
0.80
-0.740
0.4
2.08
1.882
0.80
-0.842
1.94
1.456
0.80
-0.749
0.3
2.08
1.560
0.80
-O.863
1.94
1.229
0.80
-0.774
0.25
2.08
1.362
0.87
-0.889
1.94
1.108
0.80
-0.794
O.15
2.14
0.905
0.87
-0.972
1.94
0.824
0.87
-0.861
0.1
2.14
0.632
0.94
-1.041
1.94
0.665
0.87
-O.909
0.05
5.43
0.331
0.94
-1.130
2.00
0.499
0.94
-0.971
0.4
1
6.65
2.703
5.40
-2.703
1.70
I.850
2.80
-1.740
0.9
1-75
2.323
2.90
-2.722
1.70
1.859
2.80
-1.407
0.75
1-75
2.343
2.90
-2.111
1.75
1.903
2.80
-0.856
0.6
1.80
2. 440
2.95
-1.131
1.75
2.OO7
0.70
-O.790
0.5
1.85
2.575
0.70
-0.905
1.80
2.124
0.70
-O.790
0.4
1.90
2.796
0.70
-0.905
1.85
2.179
0.70
-0.791
A- 57
TABLE A. 7 CONTINUED
8 -
0
8
- 0.10
V
i
Sa&x
putaax
^min
p%in
t
max
PW
^mln
pumin
uo
*0
*1
*0
*1
y 0
*1
y0
0.5
1.95
2.864
0.70
-0.907
1.90
1.920
0.70
-0.815
0.25
1.95
2.608
0.75
-0.928
1.90
1.757
0.75
-0.847
0.15
2.00
1.918
0.80
-1.048
1.90
1.360
0.80
-0.957
0.1
2.00
1.424
O.85
-1.172
1.95
1.118
O.85
-1.045
0.05
2.05
0.798
0.95
-1.365
1.95
0.837
0.90
-1.165
0.5
1
5-52
3.235
4.52
-3.235
1.52
1.930
2.48
-2.083
0.9
3-52
2.566
2.52
-3.257
0.8
1.56
2.473
2.52
-3.321
0.75
1.56
2.474
2.56
-3.265
1.56
1.974
2.48
-1.399
0.7
1.56
2.483
2.56
-2.934
0.6
1.60
2.546
2.56
-2.192
0.5
1.64
2.693
2.60
-1.326
1.68
2.255
0.60
-0.794
0.4
1.72
2.988
0.64
-0.919
0.3
1.80
3.500
0.64
-0.919
0.25
1.34
3>90
0.64
-0.924
1.80
2.238
0.68
-0.844
0.2
1.88
3.234
0.68
-0.961
0.15
1.92
2.862
0.72
-1.050
1.88
1.846
0.76
-0.989
0.1
1.96
2.282
0.80
-1.226
1.92
1.552
0.80
-1.120
0.05
2.00
1.372
0.88
-1.540
1.92
1.202
0.84
-1.312
0.7
1
5.64
2.814
2.08
-2.814
1.28
1.664
2.08
-1.844
0.9
2.80
2.233
2.08
-2.832
1.28
1.664
2.12
-1.842
0.75
1.28
2.153
4.96
-2.848
1.32
1.706
2.12
-1.391
0.6
1.32
2.241
2.16
-2.066
1.40
1.862
2.20
-0.864
0.5
1.40
2.444
3.64
-1.455
1.44
2.085
0.50
-0.750
0.4
1.48
2.874
O.52
-0.872
1.52
2.421
0.50
-O.751
0.3
1.64
3.628
O.52
-0.872
1.64
2.435
0.56
-0.784
0.25
1.68
3.681
0.56
-0.891
1.68
2.436
O.56
-0.837
O.15
1.84
3.685
0.64
-1.114
1.80
2.298
0.68
-1.069
0.1
1.92
3.295
O.76
-1.410
1.84
2.085
0.78
-1.278
0.05
2.00
2.167
0.88
-1.962
1.88
1.700
0.80
-1.586
1
1
1.00
1.274
1.60
-0.764
1.04
0.976
0.40
-0.653
0.90
1,02
1.289
0.40
-O.76I
1.06
0.991
0.40
-0.653
0.75
1.08
1.412
0.40
-0.761
1.16
1.116
0.40
-0.653
0.6
1.24
1.854
0.40
-0.761
1.30
1.354
0.40
-0.659
0.5
1.38
2.116
0.42
-0.777
1.40
1.596
0.42
-0.690
TABLE A. 7 CONTINUED
P =
0
P
= 0.10
V
u
JL
t
max
pu
max
PUmin
pu
max
EEH
uo
*1
yo
*1
•
yo
*1
yo
*1
«
yo
0.4
1*54
2.766
0.46
-O.85O
1.52
1-953
0.48
-0.774
0.3
1.70
3.698
O.54
-1.042
1.62
2.324
0.54
-0.941
0.25
1.78
4.121
0.60
-1.220
1.68
2.458
0.60
-1.073
0.15
1.92
4.191
O.74
“1.873
1.78
2.510
0.68
-1.470
0.1
1.96
3.509
0.82
-2. 4l4
1.82
2.378
0.74
-1.753
1.25
1
0.80
0.815
2.00
-0.760
0.83
0.576
O.34
-0.580
0.9
0.82
0.823
0.34
-0.676
0.86
0.499
0.34
-0.584
0.75
0.86
0.746
0.34
-0.680
1.23
0.631
0.37
-O.618
0.6
1.20
0.896
0.38
-0.731
1.4l
1.206
0.42
-O.707
0.5
1.42
1.746
0.43
-0.828
1.50
I.696
0.46
-0.820
0.4
1.60
3.080
0.50
-0.028
1.58
2.177
0.53
-1.000
0.3
1.76
4.436
0.59
-1.414
1.66
2.548
0.59
-1.270
0.25
1.82
4.875
0.66
-1.714
1.71
2.662
0.62
-1.446
0.15
1.94
4.636
0.78
-2.635
1.78
2.695
0.70
-1.895
0.1
1.97
3.730
O.85
-3.305
1.81
2.606
0.75
-2.176
0.05
2.00
2.200
0.93
-4.155
1.84
2.456
0.78
-2.499
1-5
1
1.26
0.702
O.29
-0.603
1.21
0.472
0.29
-0.518
0.9
1.26
O.709
O.29
-0.603
1.20
0.397
0.29
-O.523
O.75
1.27
0.546
O.30
-0.612
1.20
0.251
0.32
-0.556
0.6
1-25
0.247
0.34
-0.675
1.30
0.413
O.37
-0.648
0.5
1.25
0.439
0.38
-0.788
1.43
I.090
0.42
-0.772
0.4
1.52
2.252
O.47
-1.025
1-52
1.821
0.49
-0.973
0.3
1.72
4.440
0.56
-1.496
1.62
2.447
0.56
-1.284
O.25
1.80
5.265
0.64
-1.870
1.66
2.676
0.60
-1.489
0.15
1.92
5.446
0.77
-3.024
1.74
2.903
0.68
-2.013
0.1
1.97
4.495
0.84
-3.865
1.78
2.885
0.72
-2.341
0.05
1.99
2.695
0.92
-4.932
1.81
2.777
0.77
-2.716
2
1
1.00
0.637
1.78
-0.495
1.03
0.411
0.22
-0.424
0.75
1.05
O.671
0.23
-O.493
1.13
O.316
0.25
-O.459
0.5
1.24
O.639
1.29
-0.601
1.34
0.399
0.36
-0.690
0.4
1.35
0.612
O.36
-0.799
1.45
1.026
0.43
-0.925
0.25
1.68
4.443
0.55
-1.740
1.80
3-218
0.76
-3.491
0.15
1.88
7.070
0.72
-3.310
1.68
3.127
0.64
-2.190
2.5
1
1.17
O.438
0.18
-0.4l6
1.12
0.265
0.18
-0.357
0.9
1.17
-.394
0.18
-0.417
1.11
0.219
0.18
-O.361
0.75
0.82
0.233
0.20
-0.438
1.10
0.125
0.21
-0.390
0.6
0.92
0.089
0.25
-O.529
1.18
0.139
0.26
-0.484
0.5
1.27
0.573
0.30
-0.694
1.31
0.472
0.32
-O.625
0.4
1.44
1.205
0.40
-1.075
1.41
0.778
0.40
-O.883
A- 59
TABUS A. 7 CONTINUED
P -
0
P
- 0.10
V
i.
t
max
puinax
t ,
min
pumin
t
max
pu
max
*min
pumin
uo
*1 '
*0
*1
*0
*1
y0
*1
K
0.3
1.63
3.891
0.53
-1.892
1.50
1.704
0.48
-1.302
0.25
1.74
6.397
0.60
-2.56O
1.55
2.314
O.52
-1.586
0.15
1.90
8.491
0.75
-4.646
1.64
3.207
0.6l
-2.316
O.l
1.96
7.381
O.83
-6.168
1.69
3.466
0.66
-2.769
0.05
1-99
4.570
0.91
-8.O83
1.73
3.596
0.70
-3.284
3
1
1.00
0.425
I.85
-0.359
I.03
0,247
0.16
-0.308
0.9
1.01
0.430
0.16
-0.358
1.04
0.210
0.16
-O.312
0.75
1.03
0.369
0.16
-0.363
1.06
0.136
0.18
-0.339
0.6
1.09
0.270
0.19
-0.409
1.16
0.076
O.23
-0.429
0.5
1.18
0.196
0.23
-O.511
1.27
0.202
0.29
-0.575
0.4
1.28
0.290
0.32
-O.793
1.38
0.833
0.37
-0.849
0.3
1.51
1.800
0.46
-1.550
1.48
1.344
0.46
-1.304
0.25
1.64
4.287
0.53
-2.261
1.52
2.078
0.50
-1.614
0.15
I.87
9.831
O.71
-4.747
1.61
3.217
0.59
-2.409
0.1
1.94
9.375
0.80
-6.704
1.65
3*584
O.63
-2.901
0.05
1.99
6.147
0.90
-0.266
1.69
3.812
0.67
-3.460
4
1
1.00
O.318
1.89
-0.281
. 1.03
0.175
0.12
-0.242
0.9
1.01
O.322
0.12
-0.280
1.03
0.146
0.12
*0.244
0.75
1.02
0.244
0.13
-0.288
1.04
0.086
0.14
-0.267
0.6
1.07
0.l4l
0.16
-O.338
0
0
0.19
-0.351
0.5
0
0
0.20
-O.452
1.23
O.168
0.25
-0.500
0.4
1.30
O.569
0.30
-0.802
1.34
0.578
0.33
-0.797
0.3
1.52
2.953
0.44
-1.813
1.43
1.455
0.42
-1.307
0.25
1.61
3.973
0.53
-4.411
1.48
1.820
0.46
-1.652
0.15
1.86
12.54
0.71
-6.185
1.57
3.144
0.55
-2.539
0.1
1.94
12.27
0.81
-8.847
1.6l
3.676
0.59
-3.083
0.05
1.99
8.137
0.90
-12.32
1.65
4.059
0.64
-3.696
5
1
1.00
0.255
1.90
-0.231
1.02
0.135
0.10
-0.198
0.75
1.02
O.178
0.10
-0.239
1.03
0.062
0.12
-0.221
0.5
0
0
0.18
-0.411
1,20
0.118
0.22
-0.446
0.4
1.26
0.167
0.28
-0.818
1.31
0.578
0.30
-0.759
0,25
1.61
0.515
0.52
-3.302
1.45
I.863
0.44
-1.675
0.15
1.86
1.525
0.71
-7.620
1.53
3.033
O.52
-2.6l8
7
1
1.00
0.182
1.93
-O.170
1.02
0.094
0.07
-0.146
0.9
1.00
0.172
0.07
-0.169
1.02
0.077
0.07
-0.148
0.75
1.01
0.113
0.08
-0.179
1.03
0.040
0.08
-O.163
0.6
1.04
0.022
0.10
-0.227
0
0
O.13
-0.230
0.5
0
0
0.15
-0.355
1.17
O.O65
0.19
-0.375
0.4
1.26
0.390
0.26
-0.866
1.28
0.518
0.2?
-O.709
a-6o
TABLE A.7 CONTINUED
p =
0
P
= 0.10
V
JL
t
max
Pumax
t .
min
pumin
t
max
pu
max
pumin
uo
*1
y0
*1
“V
*1
*0
*1
*0
0.3
1.49
3.750
0.42
-2.606
1.37
1.269
O.36
-1.298
0.25
1.6l
7.398
0.52
-4.346
1.41
1.762
0.40
-1.696
0.15
1.85
20.63
0.71
-10.49
1.49
2.883
0.49
-2.705
0.1
1.94
20.95
0.80
-15.28
1.53
3.610
0.53
-3.315
0.05
1.98
14.11
0.90
-21.48
1.57
4.253
0.57
-3.998
10
1
1.00
0.127
1.95
-0.121
1.02
0.065
0.05
-0.104
0.75
1.01
0.070
0.06
-0.130
1.02
0.025
0.06
-0.118
0.5
0
0
0.13
-O.303
1.15
0.053
0.16
-0.313
0.4
1.25
0.424
0.25
-0.948
1.24
0.504
0.24
-0.663
0.25
1.61
1.002
0.51
-5.920
1.37
1.718
0.37
-1.706
0.15
1.85
2.868
0.70
-14.80
1.46
2.844
0.46
-2.757
TABLE A. 8 VALUES OF MAXIMUM AND MINIMUM PSEUDO-VELOCITIES AND ASSOCIATED TIMES
Elasto-Plastic Systems Subjected, to Ground Motion Shown:
y ‘
•
h
r\
k /\
■wfc
i
\ \
X
,*X,V
£
tN
0 =
0
p = 0.10
V
u
_z
t
max
pu
* max
^min
pUmin
t
max
pumax
^min
pUmin
uo
*1
y0
K
yo
0.1
1
16.6
0.256
1.00
-0.394
6.40
0.192
1.00
-0.371
0.9
6.6o
0.181
1.00
-0.394
2.00
0.166
1.00
-0.371
0.75
2.00
0.122
1.00
-0.395
2.00
0.149
1.00
-0.372
0.6
2.00
0.100
1.00
-0.397
2.00
0.130
1.00
-0.374
0.5
7.40
0.094
1.00
-0.399
2.00
0.116
1.00
-0.376
0.4
2.00
0.068
1.00
-0.401
2.00
0.102
1.00
-0.378
0.3
2.00
0.050
1.00
-0.405
2.00
0.087
1.00
-O.38I
0.25
2.00
0.041
1.00
-0.407
2.00
0.079
1.00
-O.383
0.15
2.00-
0.025
1.00
-0.410
2.00
0.066
1.00
-0.387
0.1
2.00
0.017
1.00
-0.413
2.00
0.059
1.00
-O.389
0.05
2.00
0.008
1.00
-0: 4l6
2.00
0.051
1.00
-0.392
0.2
1
2.10
0.904
0.90
-0.681
2.00
0.831
0.90
-0.616
0.9
2.10
0.907
0.90
-0.681
2.00
0.832
0.90
-0.6l6
0.75
2.10
0.912
0.90
-0.681
2.00
0.836
0.90
-0.6l6
0.6
2.10
0.769
0.90
-0.685
2.00
0.734
0.90
-0.619
0.5
2.00
0.660
0.90
-0.693
2.00
0.651
0.90
-0.627
0.4
2.00
0.547
0.9c
-0.704
2.00
0.560
0.90
-O.638
0.3
2.00
0.426
0.90
-0.720
2.00
0.466
0.90
-0.652
0.25
2.00
0.359
0.90
-0.731
2.00
0.418
0.90
-0.662
0.15
2.00
0.221
1.00
-0.763
2.00
O.305
0.90
-0.685
0.1
2.00
0.150
1.00
-0.783
2.00
0.268
0.90
-0.699
0.05
2.00
0.073
1.00
-0.809
2.00
0.215
0.90
-0.716
0.3
1
8.17
1.853
6.50
-1.853
1.88
1.497
3.00
-1.435
0.9
1.94
1.809
3.15
-1.600
1.88
1.502
2.95
-1.208
0.75
1.94
1.839
3.08
-1.021
1.88
1.525
2.95
-0.828
0.6
2.00
1.899
0.80
-0.840
1.94
1.571
0.80
-0.740
0.5
2.00
1.971
0.80
-0.840
1.94
1.629
0.80
-0.740
0.4
2.01
1.877
0.80
-0.843
1.94
1.456
0.80
-0.749
0.3
2.01
1.553
0.80
-0.863
1.94
1.229
0.80
-0.774
0.25
2.01
1.355
0.87
-0.890
1.94
1.108
0.80
-0.794
0.15
2.01
O.898
0.87
-0.973
1.94
0.824
0.87
-0.861
0.1
2.01
0.627
0.94
-1.042
1.94
0.665
0.87
-0.909
0.05
2.00
0.328
0.94
-1.130
2.00
0.499
0.94
-0.971
A-62
TABLE A.8 CONTINUED
P =
0
0
= 0.10
V
u
JL
t
max
pu
* max
*min
pumin
t
max
pu
* max
Snin
pUmin
u
0
*1
6 '
y0
~fT
*1
"TT
*1
0.4
1
4.00
3.724
5.25
-3-724
3.90
2.088
2.75
-2.501
0.9
4.00
3.378
2.80
-3.528
1.70
1.850
2.75
-2.523
0.75
1.75
2.323
2.85
-3.6o4
1.70
1.850
2.80
-2.571
0.6
1.75
2.326
2.90
-3.585
1.70
1.880
2.80
-I.966
0.5
1.75
2.370
2.90
-2.778
1.75
1.949
2.80
-1.443
0.4
1.80
2.499
2.90
-1.785
1.80
2.074
2.85
-0.829
0.3
1.90
2.759
0.70
-0.905
I.85
2.191
0.70
-0.791
0.25
1.95
2.958
0.70
-0.905
I.85
2.024
O.70
-0.803
0.15
1.95
2.348
0.75
-O.963
1.90
1.589
0.75
-0.688
o.l
2.00
1.812
■ 0.80
■1.071
1.95
1.292
0.80
-0.979
0.05
2.00
1 052
O.90
-1.284
1.95
O.938
O.85
-1.119
0.5
1
5-52
4.853
6.52
-4.853
3-48
2.725
2.52
-2.797
0.9
3-52
4.886
2.52
-4.071
3.48
2.218
2.52
-2.814
0-75
3-52
4.027
2.56
-4.101
1.52
1.930
2.56
-2.908
0.6
1.56
2.473
2.60
-4.312
1.56
1.949
2.60
-2.573
0.5
1.56
2.474
2.64
-4. 511
1.60
2.024
2.64
-2.067
0.4
1.60
2.546
2.68
-3-542
1.64
2.192
2.68
-1.441
0.3
1.68
2.819
2.76
-2.277
1.72
2.488
0.60
-0.794
0.25
1.72
3.087
2.80
-1.407
1.76
2.463
0.64
-0.800
0.15
1.88
3.380
0.68
-0.935
1.84
2.077
0.68
-0.897
0.1
1.92
2.862
0.72
-1.050
1.88
1.762
0.76
-1.025
0.05
2.00
1.876
0.84
-1.362
1.92
1.323
0.84
-1.238
0.7
1
2.92
2.878
2.12
-2.878
3.04
1.742
2.16
-1.951
0.9
•2.92
2.356
2.12
-2.899
1.28
1.664
2.16
-1.971
0.75
1.28
2.153
2.16
-3.066
1.32
1.683
2.24
-1.730
0.6
1.32
2.227
2.24
-2.441
1.36
1.808
2.32
-1.405
0.5
l.4o
2.408
2.36
-2.120
1.44
2.004
2.44
-1.177
0.4
1.48
2.822
2.50
-1.721
1.52
2.352
2.52
-0.835
0.3
I.60
3.624
0.52
-0.872
1.60
2.427
0.52
-0.774
0.25
1.68
3.677
0.52
-0.886
1.68
2.432
O.56
-0.821
0.15
1.84
3.696
0.64
-1.101
1.80
2.322
0.68
-1.040
0.1
1.92
3.328
0.72
-1.392
1.84
2.112
O.72
-1.252
0.05
2.00
2.201
0.84
-1.946
1.88
1.727
0.80
-1.565
1
1
1.00
1.274
3.50
-1.274
1.04
0.976
0.40
-0.653
0.9
1.02
1.289
3-50
-1.001
1.06
0.991
0.40
-0.653
0.75
1.08
1.412
0.40
-0.761
1.16
1.116
0.40
-0.653
0.6
1.24
1.855
o.4o
-0.761
1.30
1.354
0.40
-0.659
0.5
1.58
2.116
0.42
-0.777
1.40
1.597
0.42
-0.690
0.4
1.54
2.766
2.56
-1.171
1.52
1.953
2.52
-0.855
A- 6j
TABLE A. 8 CONTINUED
0
= 0
0
= 0.10
V
i
t
max
^max
"Suln
^Snin
t
max
pa
max
^min
pumin
uo
y0
TT
*1
-*T
*1
K
0.3
1.70
3.698
0.54
-1.042
1.62
2.324
0.54
-0.941
0.25
1.78
4.121
0.60
-1.220
1.68
2.458
0.60
-1.073
0.15
1.92
4.191
0.74
-1.873
1.78
2.510
0.68
-1.470
0.1
1.96
3.509
0.82
-2.414
1.82
2.378
0.74
-1.752
0.05
2.00
2.144
0.90
-3.160
1.86
2.128
0.80
-2.101
1.25
1
0.80
0.815
2.00
-0.760
0.83
0.576
0.34
-0.580
0.9
0.82
0.814
0.34
-0.676
0.86
0.499
0.34
-0.585
0.75
0.86
0.746
0.34
-0.680
1.23
0.631
0.37
-0,6l8
0.6
1.20
0.896
O.38
-0.732
1.41
1.206
0.42
-0.707
o.5
1.42
1.746
0.43
-0.828
1.50
I.696
2.50
-0.917
0.4
1.60
3.080
2.64
-1.511
1.58
2.177
2.59
-1.067
0.30
1.76
4.436
0.59
-1.414
1.66
2.548
0.59
-1.270
0.25
1.82
4.875
0.66
-1.714
1.71
2.662
0.62
-1.446
0.15
1.94
4.636
0.78
-2.635
1.78
2.695
0.70
-1.895
1.5
1
1.26
0.702
2.24
-0.837
1.21
0.472
0.29
-0.518
0.9
1.26
0.702
2.24
-0.844
1.20
0.397
2.17
-0.566
0.75
1.26
0.710
2.26
-0.698
1.20
0.251
2.21
-0.633
0.6
r.27
0.484
2.26
-O.676
1.30
0.412
0.37
-0.648
0.5
1.25
0.240
2.27
-0.825
1.43
I.O89
2.42
-0.775
0.4
1.33
0.767
0.40
-0.827
1.52
1.821
2.53
-1.088
0.3
1.61
3.197
2.65
-1.815
1.62
2.447
2.63
-1.293
0.25
1.73
4.475
0.58
-1.508
1.66
2.676
0.60
-1.488
0.15
1.90
5.657
0.73
-2.625
1.74
2.903
0.68
-2.013
2
1
1.00
0.637
3-25
-0.637
I.03
0.411
0.22
-0.424
0.9
1.01
0.645
3.25
-0.500
1.05
0.359
0.23
-0.428
0.75
1.05
0.671
0.23
-0.493
1.13
0.316
0.25
-0.459
0,6
1.13
0.591
0.25
-0.524
1.26
0.360
0.31
-0.554
0.5
1.24
0.639
0.29
-0.601
1.34
0.399
2.35
-0.722
0.4
1.35
0.612
2.36
-0.848
1.45
1.025
2.45
-0.992
0.3
1.54
2.169
2.57
-2.133
1.55
2.097
2.55
-1.394
0.25
1,68
4,443
2.70
-2.241
I.60
2.539
2.60
-1.580
0.15
1.88
7.070
0.72
-3.3U
1.68
3.127
0.64
-2.190
0,1
1.95
6.463
0.81
-4.559
1.72
3.256
0.68
-2.590
0.05
1.99
4.149
0.90
-6.214
1.76
3.272
0.73
-3.046
TABLE A. 8 CONTINUED
P
= 0
P
= 0.10
*1'
u
JL
t
max
pu
* max
^min
pumin
t
max
pu
max
^min
pumin
uo
*1
y0
*1
*0
"l"
2.5
1
1.17
0.458
2.16
-0.472
1.12
0.265
0.18
-0.557
0.9
1.17
0.458
2.16
-0 . 449
1.11
0.219
0.18
-O.56I
0.75
l.lfl
0.500
2.16
-0.445
1.10
0.125
2.08
-0.597
0.6
0.87
0.152
0.25
-0. 488
1.18
0.159
0.26
-0.484
0.5
1.18
0.272
0.28
-0.615
1.51
0.472
0.52
-0.625
0.4
1.40
1.164
0.57
-0.921
1.41
0.778
0.4o
-0.885
0.5
1-57
2.568
2.60
-2.840
1.50
1.704
2.50
-1.410
0.25
1.70
5.496
2.72
-2.676
1.55
2.514
2.55
-1.661
0.15
1.89
8.517
0.74
-4.541
1.64
5.207
0.61
-2.516
a-6?
TABLE A. 9 VALUES OF MAXIMUM AND MINIMUM PSEUDO-VELOCITIES AND ASSOCIATED TIMES
Elasto-Plastic Systems Subjected to Ground Motion Shown:
-*t
y *
tv<vV
-►t
t
■ VVVV
h» 4* *4 M
0 =
0
P
= 0.10
V
u
_z
t
max
pumax
^min
pUmin
t
max
pu
max
Snin
pumin
uo
*1
"77
*1
*1
“JT
*1
yo
0.1
1
14.60
0.413
19.60
-0.413
4.20
0.364
1.00
-0.371
0.9
4.40
0.379
1.00
-0.394
4.00
0.316
1.00
-O.37I
0.75
4.20
0.276
1.00
-0.395
4.00
0.243
1.00
-0.372
0.6
4.00
0.181
1.00
-0.397
4.00
0.207
1.00
-0*374
0.5
4.40
0.172
1.00
-0.398
4.00
0.200
1.00
-0.376
0.4
4.40
0.145
1.00
-0.401
4.00
0.176
1.00
-0.378
0.3
4.40
0.108
1.00
-0.4o4
4.00
0.151
1.00
-O.38I
0.25
4.40
0.088
1.00
-0.4o6
4.00
0.138
1.00
-0.383
0.15
4.20
0.054
1.00
-0.410
4.00
0.117
1.00
-0.387
0.1
4.20
0.038
1.00
-0:412
4.00
0.104
1.00
-O.389
0.05
4.00
0.016
1.00
-0.416
4.00
0.091
1.00
-0.392
0.2
1
2.10
0.904
0.90
-0.682
2.00
0.831
0.90
-0.6l6
0.9
2.10
O.907
0.90
-0.681
2.00
O.832
0.90
-0.616
0.75
2.10
0.912
0.90
-0.681
2.00
O.836
0.90
-0.6l6
0.6
4.00
0.832
0.90
-0.685
4.00
0.798
0.90
-0.619
0.5
4.00
0.859
0.90
-0.693
4.00
0.839
0.90
-0.627
0.4
4.10
0.861
0.90
-0.704
4.00
0.762
0.90
-O.638
0-3
4.10
0.725
0.90
-0.720
4.00
0.661
0.90
-0.652
0.25
4.10
0.632
0.90
-0.731
4.00
0.607
0.90
-0.662
0.15
4.10
0.414
1.00
-0.763
4.00
0.480
0.90
-0.685
0.1
4.20
0.291
1.00
-0.783
4.00
0.415
0.90
-0.699
0.05
4.10
0.144
1.00
-0.809
4.00
0.341
0.90
-0.716
0.3
1
1.94
1.805
3.08
-1.805
1.88
1.497
3.00
-1.435
0.9
1.94
1.811
3.03
-1.483
1.88
1.502
2.95
-1.208
0.75
1.94
1.845
3.02
-O.927
1.88
1.525
2.95
-0.828
0.6
2.00
1.908
0.80
-0.840
1.94
1.571
0.80
-0.740
0.5
4.00
2.066
0.80
-0.840
3.95
1.701
0.80
-0.740
0.4
3-95
2.149
0.80
-0.844
3.95
1.601
0.80
-0.749
0.3
3-95
1.969
0.80
-0.866
3.95
1.431
0.80
-0.774
0.25
4.00
1.818
0.87
-0.894
3.95
1-335
0.80
-0.794
0.15
4.00
1.381
0.87
-0.977
3.95
1.071
0.87
-0.861
0.1
4.00
1.040
0.94
-1.046
3.95
0.901
0.87
-0.909
0.05
4.02
0.591
0.94
-1.132
3.95
0.702
0.94
-0.971
A-66
TABUS A. 9 CONTINUED
6
m
P
= 0.10
V
u
JL
t
max
pu
* max
^min
Pumin
wm
pu
max
BHWI
pumin
uo
“37
*1
ti
^7
o.4
1
8.90
4.369
7.65
-4.369
3.80
2.755
2.75
-2.501
0.9
3.90
4.333
5.15
-3.579
3.80
2.728
2.75
-2.501
0.75
3-95
3.884
2.80
-3.532
3.80
1.982
2.75
-2.529
0.6
3-95
2.351
2.85
-3.647
1.70
1.860
2.80
-2.248
0.5
1.75
2.328
2.90
-3.486
1.75
1.906
2.80
-1.722
0.4
1.75
2-397
2.90
-2.499
1.75
2.010
2.80
-1.098
0.5
1.85
2.605
2.90
-1.220
1.85
2.202
0.70
-0.790
0.25
1.90
2.785
0.70
-O.905
3.85
2.161
0.70
-0.795
0.15
3.90
2.941
0.75
-0.933
3.90
1.833
0.75
-O.867
0.1
3.95
2.618
0.80
-1.025
3.90
1.578
0.80
-0.958
0.05
4.00
1.871
0.90
-1.237
3.95
1.223
0.85
-1.103
0.5
1
7-52
6.470
6.52
-6.470
3.50
3.436
4.48
-3.194
0.9
3-52
5.688
4.52
-6.514
3.52
3.454
2.52
-2.797
0.8
3-52
5-710
4.52
-5.489
0.75
3.56
5-759
4.52
-4.773
3.56
3.128
2.52
-2.808
0.7
3.56
5.838
2.52
-4.071
0.6
3.60
5.652
2.52
-4.073
3.60
2.161
2.56
-2.923
0.5
3-64
4.269
2.56
-4.185
1.56
1.943
2.60
-2.640
0.4
3.68
2.629
2.64
-4.509
1.60
2.035
2.64
-2.018
0.3
1.60
2.546
2.68
-3.543
1.68
2.265
2.72
-1.215
0.25
1.64
2.693
2.72
-2.749
1.72
2.463
0.60
-0.794
0.2
1.72
2.988
2.80
-1.717
0.15
1.80
3.500
0.64
-0.919
3.80
2.274
0.68
-0.846
0.1
3.88
3.516
0.68
-0.961
3.84
2.056
0.72
-0.958
0.05
3.92
3.036
0.80
-1.226
3.88
1.657
0.80
-1.181
0.03
3.96
2.418
0.88
-1.462
0.01
4.00
1.111
0.96
-1.832
0.7
1
2.92
2.869
2.12
-2.869
3.04
1.755
2.16
-1.951
0.9
2.92
2.356
2.12
-2.899
1.28'
1.664
2.16
-1.971
0.75
1.28
2.153
2.16
-3.066
1-32
1.683
2.24
-1.730
6.6
1.32
2.227
2.24
-2.441
1.36
I.808
2.32
-1.405
0.5
1.40
2.408
2.36
-2.120
1.44
2.004
2.44
-1.177
0.4
3-52
2.885
2.50
-1.721
3.52
2.355
2.52
-0.835
0.3
1.60
3.624
0.52
-0.872
1.60
2.427
0.52
-O.774
0.25
1.68
3.677
0.52
-O.887
1.68
2.432
O.56
-0.821
0.15
3.84
3.981
0.64
-1.101
3.76
2.360
0.68
-1.040
0.1
3.88
U.133
O.72
-1.392
3.84
2.225
0.72
-1.252
0.05
3.96
3.330
0.84
-1 ,946
3.88
1.917
0.80
-1.565
TABLE A. 9 CONTINUED
0 *
0
0
- 0.10
V
Hi
t
max
^Umax
^min
Pumin
t
max
^min
^Umln
uo
*1
*0
*1
K
*1
K
1
i
1.00
1.274
3.60
-0.767
1.04
0.976
0.40
-0.653
0.9
3.02
1.289
0.40
-O.76I
1.06
0.991
0.40
-0.653
0.75
1*08
1.412
0.40
-O.761
1.16
1.116
o.4o
-0.653
0.6
1.24
1.855
0.40
-O.76I
1.30
1.354
0.40
-0.659
o.5
1.38
2.116
0.42
-0.777
1.40
1.597
0.42
-0.690
0.4
3-58
2.862
2.56
-1.171
3.54
I.966
2.52
-0.855
0.3
1.70
3.698
0.54
-1.042
1.62
2.324
0.54
-0.941
0.25
3.78
4.190
0.60
-1.220
1.68
2.458
0.60
-1.073
0.15
3.88
5.170
0.74
-1.873
3.76
2.552
0.68
-1.470
0.1
3-94
4.963
0.82
-2.414
3.80
2.465
0.74
-1.752
0.05
3.98
3.551
0.90
-3.160
3.86
2.254
0.80
-2,101
1.5
l
3-22
0.990
2.24
-0.837
3-15
0.534
0.29
-0.518
0.9
3.23
0.999
2.23
-0.838
3.16
0.461
2.16
-0.548
0.75
3.24
0.809
2.24
-0.847
1.20
0.277
2.16
-0.644
0.6
3*25
0.712
2.26
-0.642
1.27
0.306
0.36
-0.632
0.5
3.25
0.484
2.26
-0.683
1.40
0.976
0.42
-O.JkQ
0.4
0.78
0.198
2.27
-0.84],
3.52
1.743
2.52
-1.058
0.3
3.50
1.958
2.50
-1.263
1.61
2.396
2.62
-1.276
0.25
3.66
3.358
2.65
-1.798
1.66
2.646
0.60
-1.454
0.15
3.83
6.309
0.68
-2.241
3.73
2.917
0.67
-1.984
2
1
1.00
0.637
3.78
-0.496
I.03
0.411
0.22
-0.424
0.9
3.01
0.645
0.23
-0.493
1.05
0.359
0.23
-0.428
0.75
1.05
0.671
0.23
-0.493
1.13
0.315
0.25
-0.459
0.6
1.13
0.591
0.25
-0.524
1.26
0.360
0.31
-0.554
0.5
1.24
0.639
0.29
-0.601
1.34
0.399
2.35
-0.722
0.4
1.35
0.612
2.36
-0.848
3-45
1.032
2.45
-0.992
0.3
3.58
2.509
2.57
-2.133
3.55
2.102
2.55
-1.394
0.25
1.68
4.443
2.70
-2.241
1.60
2.539
2.60
-1.588
0.15
3.85
8.363
0.72
-3.311
3.68
3.129
0.64
-2.189
0.1
3.92
8.951
0.81
-4.559
3-72
3.264
0.68
-2.590
0.05
3-97
6.849
0.90
-6.214
3.76
3.290
0.73
-3.046
2.5
1
3.15
0.515
2.16
-0.472
3.08
0.281
0.18
-0.357
0.9
3.15
0.500
2.16
-0.473
3.08
0.242
0.18
-0.361
0.75
3.15
0.387
2.16
-0.432
3.09
0.152
2.08
-0.397
0.6
3.15
0.239
0.21
-0.455
3.18
0.142
0.26
-0.484
0.5
3.15
0.125
0.26
-0.543
1.31
0.472
0.32
-0.625
A-68
TABLE A. 9 CONTINUED
V i
uo
6
■ 0
0
= 0.10
t
max
h
pumax
"^min
*1
pUmin
t
max
*1
pumax
y0
m
pumin
^0
0.4
1-33
0.837
0.34
-0.777
3.41
0.778
0.40
0.3
3-50
1.184
2.49
-2.893
3-50
1.709
2.50
-1.410
0.25
3.68
4.290
2.67
-3.018
3-55
2.317
2.55
-1.661
0.15
3 .84
9.830
0.71
-3.986
3.64
3.207
0.61
-2.316
TABLE A. 10 VALUES OF MAXIMUM AND MINIMUM DEFORMATIONS
AND DISPLACEMENTS WITH THE ASSOCIATE TIMES
Elasto-Plaatic System, Damping Factor, p = 0.02, Eureka Earthquake
yQ ■ 10,00 in., yQ * 12.50 in. /sec., yQ * 0.178 g, Duration of Quake ■ 20 sec.
u
JL
u
0
Deformations
Absolute Displacements
t
max
sec .
u
max
in.
^min
sec .
umin
in.
t
max
sec .
xmax
in.
Snin
sec .
Xmin
in.
f - 1/25
= .04 cps
1.00
4.80
9.563
12,60
-5.783
0.80
4.80
9-571
10.22
-3.632
19-84
3-341
0.70
4.80
9.597
10.22
-2.647
19.40
4.963
0.60
17.78
9.868
9.00
-1.872
19.OO
6.622
0.50
4.80
9.674
8.64
-1.175
18.40
6.182
0.40
4.80
9*724
8.64
-0.743
0.30
4.80
9.778
8.64
-0.468
7-20
-0.746
0.25
4.80
9.812
8.64
-0.344
7.06
-0.609
0.20
4.80
9.846
2.80
-0.320
6.96
-0.479
0.15
4.80
9.879
2.80
-0.320
6.88
-0.363
0.10
4.80
9.915
12.60
-0.355
6.72
-0.255
0.05
4.86
9.957
12.60
-0.671
6.44
-0.146
f - 1/15 -
■ 0.067 cps
1.00
4.80
8.861
9.00
-7.897
15.60
6.218
8.64
-7.955
0.818
14.96
9-724
8.64
-5.920
15.34
7.519
0.716
14.78
9-885
8.64
-4.569
15.10
7.573
0.6l4
14.60
9-911
8,60
-3.289
14.96
7.518
0.511
14.56
9-237
8.60
-2.382
14.88
6.811
0.409
4.80
9-329
8.60
-1.634
14.88
5.840
0.307
4.80
9-479
8.64
-1.047
14.80
4.618
0.256
4.80
9.567
8.64
-0.760
14.78
3-987
0.205
4.80
9.656
8.64
-0.509
14.78
3.308
6.74
-0.781
0.153
4.80
9.745
8.64
-0.307
14.78
2.557
0.102
4.80
9.843
2.80
-0.270
6.60
-0.500
0.051
4.80
9.953
2.80
-0.270
6.20
-0.223
f - l/lO
* 0.1 cps
1.00
12.00
9.603
18.06
-9.6l6
12.36
9.562
17-66
-13.IO
0.80
4,60
7.688
7.44
-9.117
12.50
7.019
17-84
-12.00
0.70
4.60
7.766
7.44
-7.849
12.50
6.468
17.84
-10.34
0.60
4.70
7.911
7.44
-6.389
12.50
6.133
17.96
-8.382
0.50
4.70
8.130
7.00
-4.964
12.60
5.738
7.10
-6.727
0.40
4.80
8.427
6.96
-3.530
12.66
5-346
7.06
-5.270
0.30
4.80
8.773
8.58
-2.337
12.88
4.567
6.96
-3.874
0.25
4.80
8.982
8.58
-1.735
13.00
4.140
6.88
-3.123
0.20
4.80
9.193
8.60
-1.217
13-16
3.613
6.80
-2.420
TABLE A. 10 CONTINUED
u Deformations Absolute Displacements
u
0
t
max
u
max
"Sain
umin
^max
Xmax
Snin
Xmin
sec .
in.
sec .
in.
sec .
in.
sec .
in.
0.15
4.80
9.403
8.60
-0.779
13-38
3-007
6.72
-1.762
0.10
4.80
9.641
8.64
-0.370
13-78
2.417
6.62
-1.102
0.05
4.80
9.902
1.00
-0.213
f = 1/7 -
0.143 cps
6.32
-0.450
1.00
9.90
11.009
6.32
-11.83
10.00
11.139
6.38
-13.62
0.80
3.98
7.067
6.60
-12.06
10.12
6.380
6.44
-13.76
0.70
3.98
7.067
6.62
-12.37
10.32
3-855
6.58
-14.00
0.60
3-98
7.067
6.62
-12.78
10.40
1.374
6.72
-14.40
0.50
3-98
7.087
6.62
-H.03
10.54
O.876
6.72
-12.66
0.40
3.98
7.151
6.62
-9.090
10.60
0.6l4
6.72
-10.72
0.30
3.98
7.269
6.62
-6.970
10.72
O.585
6.78
-8.613
0.25
4.60
7.485
6.62
-5-742
10.86
0.745
6.78
-7.383
0.20
4.70
7.887
6.62
-4.282
10.98
1.111
6.78
-5-910
0.15
4.80
8-354
6.62
-2.830
11.20
1.525
6.72
-4.448
0.10
4.80
8.903
8.58
-1.484
11.46
1.816
6.72
-2.955
0.05
4.80
9.550
8.64
-0.531
13-48
1.324
6.58
-1-351
t - 1/5 '
■ 0.2 cps
-15.86
1.00
8.22
12.272
5.88
-12.81
8.32
11,56
5.76
0.80
8.22
6.697
6.12
-13.14
8.40
6.043
5.84
-15.92
0.70
3.90
6.273
6.18
-13.57
8.40
2.903
5.88
-16.07
0.60
3.90
6.273
6.18
-14,17
3.12
0.429
5.94
-16.32
0.50
3.90
6.273
6.22
-14.95
3.12
0.429
16.20
-16.97
0.40
3.90
6.307
6.22
-13.08
3.12
0.429
6.12
-14.85
0.30
3.90
6.448
6.24
-10.59
3.12
0.429
6.18
-12.32
0.25
3.96
6.569
6.24
-9.250
3,12
0.429
6.22
-IO.96
0.20
3.96
6.744
12.46
-7.987
3.12
0.429
6.32
-9.495
0.15
3.96
6.962
12.60
-7.363
3.12
0.429
16.20
-8.853
0.10
4.68
7.395
15.84
-6.916
3.12
0.429
19.80
-9.343
0.05
4.80
8.594
8.58
-2.061
3.12
0.430
19.08
-4.252
f - l/4 -
0.25 cps
1.00
7.56
13.24
5.70
-11.82
7.52
11.521
5.52
-16.35
0.80
7.58
IO.87
5.70
-H.89
7.56
9.134
5.52
-16.36
0.70
7.60
7.990
5.76
-12.08
7.58
6.248
5.52
-16.39
0.60
3.84
5.476
5.76
-12.49
7.60
3.022
5.52
-16.51
0.50
3.84
5.476
5.84
-13.17
3.00
0.413
5.60
-16.76
0.40
3.84
5.477
5.84
-13.65
3.00
0.413
5.70
-16.86
0.30
3.84
5.585
6.12
-11.46
3.00
0.413
5-76
-14.26
0.25
3.90
5-708
6.18
-10.33
3.00
0.413
5.76
-12.81
0.20
3.90
5.907
6.20
-9.033
3.00
0.413
19-14
-11.42
A-71
TABLE A. 10 CONTINUED
u
JL
Deformations
Absolute Displacements
uo
V&x
sec.
u
max
In.
^min
sec .
^min
In.
t
max
sec.
xmax
in.
^min xmin
sec. in.
0.15
5.90
6.191
15.76
-7.956
3.00
0.413
19.20 -10.49
0.10
5.96
6.588
12.60
-6.813
3.00
0.413-
18.96 -8.887
0.05
4.68
7.461
12.60
-5.046
t - 1/3 -
3.00
0.33 cps
0.413
18.84 -7.532
1.00
9-72
10.6l
8.28
-10.75
9.78
10.51
5.08 -15,64
0.80
6.84
10.78
5.28
-7.974
9*78
10.29
5.08 -15.64
0.70
6.88
10.14
5.34
-7,986
9.84
9.607
5.08 -15.64
0.60
6.90
7.908
5.34
-8.161
13.02
7-393
5.08 -15.64
0.50
13.26
5.250
5.52
-8.681
13.08
4.863
5.10 -15.68
0.40
3-72
4.352
5.56
-9.624
13.16
1.835
5.16 -15.77
0.50
3.78
4.491
5.64
-8.001
13*26
1.627
5.22 -13.49
0.25
3.84
4.643
5.76
-7.210
13.38
1.450
5.22 -12.21
0.20
3.84
4.903
5.82
-6.321
13.44
1.324
5.28 -10.77
0.15
3.84
5.248
5.82-
-5-467
2.16
0.341
5.40 -9-264
0.10
3.90
5.793
12.36
-5.088
2.16
0.341
5.56 -7.573
0.05
3.96
6.55*
12,54
-6.441
t - 1/2.5
2.16
■ 0.4 cps
0.341
18.48 -8.010
1.00
6.42
7.112
10.22
-6.694
9.00
6.591
4.80 -14,63
0.80
6.44
7.268
5.10
-5.113
9.02
7.034
4.80 -14.6*3
0.70
9.12
7-243
5.10
-5.H5
9.06
7.214
4.80 -14.63
0.60
9.18
6.112
5.10
-5.428
9-16
6.020
4.86 -14.70
0.50
9.30
4.417
5.22
-5.660
9-24
4.239
4.88 -14.34
0.40
9.38
4.171
5.46
-4.924
9.38
3,925
4.96 -12.96
0.30
3-78
4,180
5.56
-4.587
9.48
2.402
5.04 -11.36
0.25
3.78
4.422
5.60
-4.161
9.54
1.931
5.04 -10.30
0.20
3.84
4.792
5,76
-3.608
13.26
2.305
5.10 -9,014
0.15
3.90
5.244
5.82
-3.202
13.38
2.313
5.20 -7.681
0.10
3.90
5.861
6.18
-2.922
13.38
0.345
5.40 -6.252
0.05
4.62
6.731
12.50
-4.858
f - l/2
1.80
■0.5 cps
0.326
18.34 -6.188
1.00
19.02
3.228
4.38
-4.986
12.96
3.325
4.50 -13.50
0.80
3.56
3.221
4.44
-5,066
12.88
2.055
4.56 -13.74
0.70
3.56
3.221
4,44
-5.190
12.84
1.320
4.62 -l4>04
0.60
3.56
3.225
4.44
-4.805
12.78
1,151
4.68 -13.91
0.50
3.60
3.283
4.44
-3.819
12.84
3-554
4.70 -13.03
0.40
13.20
4.753
5.20
-3.069
13.02
4.683
4.80 -12.03
TABLE A 10 CONTINUED
u
JL
Deformations
Absolute Displacements
uo
t
max
sec.
umax
in.
^min
sec .
u '
min
in.
t
max
sec.
X
max
in.
Vin
sec.
xmin
in.
0.30
13.52
4.488
5.52
-2,858
13.08
3.918
4.88
-10.81
0.25
3-78
4.030
5.56
-3.001
13.14
2.738
4.96
-10.06
0,20
3.81*
4.518
5.64
-2.769
13.16
2.309
5.04
-8.871
0.15
3.81*
5.132
5.76
-2.581
13.20
1.904
5.16
-7.516
0.10
3.90
5.902
6.18
-2.619
13.26
1.026
5.40
-6.194
0.05
4.68
7.030
8.52
-2.950
f - 1/1,5 -
3.12
0.67 cps
0.419
5.76
-4.640
1.00
6.27
4.433
7.00
-4.162
10,68
3-795
4.23
-11.69
0.80
6.30
3.668
5.52
-4.024
12.27
3.311
4.23
-11.70
0.70
6.30
2.723
5.49
-4.153
10.68
2.510
4.26
-11.77
0.60
3.51
1-942
5.49
-4.287
10.68
1.760
4.28
-11.95
0.50
3.51
1.942
5.46
-4.471
IO.65
1.042
4.35
-12.28
0.40
3-51
1.947
5.43
-4.495
IO.65
0.554
4.41
-12.39
0.30
3.54
2.073
5.43
-4.105
2.79
0.512
5.04
-H.72
0.25
3.56
2.266
5.49
-3-975
2.79
0.512
5.01
-11.57
0.20
3.60
2.584
5.52
-3-942
2.79
0.5U
5.01
-11.20
0.15
3.63
3-052
5.58
-4.078
2.79
0.512
5.01
-10.58
0.10
3.81
3.853
5-79
-4.397
2.79
0.512
5.16
-9.365
0.05
3-90
5.218
6.18
-5.138 2.79
f - 1/1.25 - 0.80
0.512
5.58
-7.744
1.00
5.91
5.438
5.25
-4.949
8.43
3.383
5,20
-12.36
0.80
5.94
4.356
5.25
-4,979
8.46
2.553
5.20
-12.36
0.70
4.68
3.414
5. 28
-5.119
8.49
1.598
5.20
-12.39
0.60
4,68
3-417
5.31
-5.003
8.52
0.946
5.22
-12.16
0.50
4.70
2.983
5.31
*4,430
8.55
0.722
5.22
-11.55
0.1*0
4.71
I.698
5.34
-4.740
2.43
O.499
5.20
-U.8l
0.30
3.48
1.599
5.40
-5-564
2.43
O.499
5.20
-12.53
0.23
3.48
1.622
5.46
-5.607
2.43
0.499
5.19
-12.45
0.20
3.52
1.802
5.49
-5.422
2,43
O.499
5.19
-12.09
0.15
3.56
2.176
5.52
-4.753
2.43
0.499
5.13
-11.42
0.10
3.63
2.850
5.58
-4.216
2.43
0.499
5.10
-10.53
0.05
3.84
4.212
5.82
-4.519
f - 1
2.43
cps
0.499
5,28
-8.810
1.00
5.55
2.994
5.07
-5.306
12.60
2.501
4.98
-12.39
0.80
^.53
1.872
5.07
-3.401
12.63
1.517
4.98
-12.39
0.70
^.53
1,872
5.08
-3-533
12.63
0.958
4.98
-12.41
0.60
^•53
1.872
5.10
-3*768
3.03
0.501
5*01
-12.48
0.50
4.53
1.884
5.13
-3.530
3.03
0.501
5.01
-12.07
0.1*0
4.56
1.729
5.19
-3.238
3.03
0.501
5.04
-11.52
TABLE A . 10 CONTINUED
u Deformations Absolute Displacements
u
0
t
max
sec .
u
max
in.
t .
min
sec .
U;nin
in.
t
max
sec .
X
max
in.
^min
sec .
x .
min
in.
0.50
4.59
1.148
5.25
-3-464
3.03
0.501
5.08 -11.27
0.25
5.27
1.187
5.34
-3.973
3.03
0.501
5.10 -11.48
0.20
3-51
1.550
5.46
-4.231
3.03
0.501
5.13 -11.37
0.15
3.54
2.250
5.52
-4.289
3.03
0.502
5.13 -10.86
0.10
3.63
3.338
5,61
-4.302
3-03
0.505
5.16 -IO.09
0.05
3.84
4.894
5.82
-3.560
f * l/0.7 -
3.06
1.43 cps
0.377
5.28
-7.818
1.00
5.97
1.444
4,92
-1.168
12.54
1.216
4.88 -11.14
0.80
5-97
1.447
4,92
-1.168
12.56
1.588
4.88 -11. 14
0.70
5-97
1.121
4.96
-1.215
12.56
1.449
4.88 -11.14
0.60
4.53
0.770
5-01
-1.407
12.56
1.088
4.88 -11.17
0.50
4.53
0.771
5.06
-1.629
12.57
0.625
4.91 -11.13
0.40
4.53
0.804
5.09
-1.615
2.54
0.443
4.91 -10.85
0.30
4.53
0.777
5.15
-1.819
2.54
0.443
4.94 -10.70
0.25
3-27
0.848
6.17
-2.249
2.54
0.443
4.96 -10.62
0.20
3.51
1.333
5.30
-2.129
12.62
0.599
5.00 -10.17
0.15
3.56
2.037
5.51
-2.407
12.66
0.448
5.04
-9.674
0.10
3.63
2,931
5.60
-3.140
2.54
0.430
5.08
-9-411
0.05
3.84
4.338
5.81
-3.659
f * l/0.5
2.57
■ 2 cps
0.373
5.22
-8.177
1.00
6.83
1.317
6.59
-1.317
12.76
1.060
4.73 -IO.56
0.80
6.35
1.108
6.59
-1.231
12.75
0.850
4.73 -IO.56
0.70
6.35
1.128
6.59
-0.946
12.74
0.991
4.73 -IO.56
0.60
6.36
1.175
4.71
-0.749
12.71
1.149
4.73 -IO.56
0.50
6.36
0.974
6.09
-0.821
12.68
1.088
4.74 -IO.58
0.40
6.38
0.635
6.09
-0.941
12,65
0.905
4.76 -10.67
0.30
4.46
0.310
6.11
-1.046
12.63
0.776
4.79 -10.77
0.25
3.51
0.262
6.12
-1.093
12.63
0.727
4.82 -10.86
0.20
3.51
0.262
6.14
-1.508
2.87
0.452
4.83 -11.01
0.15
3-51
0.266
6.15
-1.888
2.87
0.452
4.88 -10.99
0.10
3.50
0.460
6.15
-2.150
2.87
0.452
4.94 -IO.56
0.05
3.63
2.512
5.58
-2.454
f - 1/0.4
12.65
■2,5 cps
0.617
5.03
-9.OO9
1.00
7*5^
O.983
7.74
-1.024
12.74
1.278
4.86 -10.28
0.80
6.74
0.936
6.54
-0.818
12,74
1.262
4.86 -10.28
0.70
6.74
0.735
6.54
-0.824
12.74
1.132
4.86 -10.28
0.60
6.33
0.604
6.56
-0.845
12.74
0.993
4.86 -10.26
0.50
6.35
0.526
6.56
-0.735
12.74
0.973
4.86 -10.32
0.40
6.33
0.446
4.07
-0.665
12.74
O.98I
4.88 -10.36
A-7*
TABLE A. 10 CONTINUED
Deformations Absolute Displacements
u
0
t
max
sec .
\ax
in.
Snin
sec .
umin
in.
t
max
sec .
xmax
in.
^min
sec .
xmin
in.
0.50
3.50
0.366
5.46
-0.773
12,74
0.857
4.77 -10.30
0.25
3.50
0.379
5.48
-0.855
12.74
0.811
4.76 -10.24
0.20
3.51
0.400
5.49
-O.927
12.74
0.705
4.77 -10.26
0.15
3.51
0.419
6.12
-1.091
12.74
0.474
4.83 -10.46
0.10
3.50
0.472
6.14
-1.848
2.73
0.408
4.89 -10.62
0.05
3.57
I.506
5.52
-2.482
f = 1/0.3 -
2.78
3.33 cps
0.429
4.98
-9.844
1.00
6.23
0.481
4.92
-0.416
12.62
1.024
4.90 -IO.33
0.80
6.23
0.446
M3
-0.417
12.62
1.088
4.90 -IO.33
0.70
6.24
0,362
M3
-0.431
12.6^
1.054
4.90 -IO.33
0.60
4.40
0.218
4.95
-0.501
12.63
0.947
4.90 -IO.35
0.50
3-78
0.182
4.98
-0.710
12.63
0.695
4.91 -10.44
0.40
3.00
0.168
6.06
-1.071
2.70
0.460
4.92 -IO.53
O.JO
3.01
0.171
5.06
-1.144
2.70
0.460
4.91 -10.59
0.25
3.45
0.235
5.06
-1.194
2.70
0.460
4.90 -10.60
0.20
3.48
0.347
5.08
-1.177
12.65
0.459
4.89 -10.55
0.15
3.50
0.481
5.46
-1.193
12.63
0.472
4.86 -10.46
0.10
3.50
0.539
5,50
-1.721
2.74
0.383
4.90 -10.40
0.05
3.60
1.884
5.5^
-2.220
f - 1/0.25
12.61
m 4 CpS
0.610
4.98
-9.582
1.00
6.64
O.291
4.88
-0.278
12.63
0.992
4.87 -10.29
0.80
6.64
0.170
6.00
-0.347
12.62
0.926
4.87 -10.29
0.70
4.74
0.166
6.01
-0.372
12.62
0.892
4.87 -IO.30
0.60
3.40
0.149
4.92
-0.356
12.62
0.898
4.88 -IO.31
0.50
3.40
0.149
M7
-0.497
12.62
O.83O
4.89 -10.36
0.40
3-41
0.158
5.02
-0.717
12.61
0.630
4.89 -10.42
O.JO
3^5
0.205
5.06
-1.013
12.61
0.438
4.90 -10.49
0.25
3^7
0.241
5.Q8
-1.241
f - 1/0.2
2.87
- 5 cps
0.374
4.90 -10.58
1.00
5.54
0.139
6.01
-0.151
12.65
0.982
4.83 -10.18
0.80
5.5^
0.118
6.01
-0.144
12.65
0.959
4.83 -10.18
0.70
5*5^
O.O850
6.01
-0.149
12.65
0.939
4.84 -IO.19
0.60
3*3^
0.0751
6.00
-0.175
12.65
O.899
4.85 -10.22
0.50
3-31*
0.0751
6.02
-0.313
12.65
0.748
4.86 -10.26
0.40
3.3^
0.0784
5.02
-0.562
12.65
0.555
4.87 -10.34
0.30
3.38
0.105
5.07
-1.021
.2.74
0.355
4.89 -10.46
0.25
3.46
0.180
5.10
-1.3^1
2.74
0.355
4.90 -10.62
0.20
3.^9
0.279
5.3^
-1.512
2.74
0.356
4.89 -IO.65
A-75
TABLE A. 10 CONTINUED
u
JL
Deformations
Absolute Displacements
uo
t
max
sec .
u
max
in.
^min
sec .
umln
in.
t
max
sec .
X
max
in.
t
min
sec .
x .
min
in.
1.00
5.85
0.0826
3.83
f = 1/0.15 *
-O.O837
6.67 cps
12.62
0.974
4.86
-10.08
0.80
3.75
0.0691
3.99
-0.0823
12.62
0.973
4.86
-10.08
0.70
3.75
0.0677
4.91
-0.0788
12.62
0.970
4.86
-10.08
0.60
3*76
0.0512
4.92
-0.0942
12.62
0.947
4.86
-10.09
0.50
3.^3
0.0364
4.93
-0.0992
12.62
0.917
4.85
-10.11
0.40
3.43
O.O367
6.51
-0.173^
12.62
0.862
4.84
-10.13
0.30
3.45
0.0451
6.53
-O.6352
12.62
0.426
4.85
-10.27
0.25
3.46
0.0523
6.07
-O.9678
2.81
0.353
4.88
-10.37
1.00
5.49
0.0415
6.46
f = 1/0.125
-0.0466
* 8 cps
12.61
O.970
4.83
-10.06
0.719
3-73
0.0291
6.48
-0.0883
12.61
0.915
4.84
-10.06
0.63
3.73
0.0210
6.48
-0.1127
12.61
0.886
4.84
-10.07
0.54
3.38
0.0206
6.49
-O.1370
12.61
0.858
4.84
-10.08
0.45
3*38
0.0206
6.50
-0.1752
12.61
0.820
4.84
-10.08
1.00
5-85
0.0225
3.64
f - l/o.l -
-0.0217
10 cps
12.61
0.968
4.84
-10.06
0,90
5.85
0.0202
4.90
-0.0227
12.61
0.968
4.84
-10.06
0.80
3-10
0.0124
6.46
-0.0293
12.61
0.957
4.84
-10.06
0.75
3.10
0.0124
6.47
-0.0406
12.61
0.945
4.84
-10.06
0.70
3*10
0.0124
6.47
-0.0632
12.61
O.921
4.84
-10.07
0.65
3.10
0.0124
6.46
-0.1088
12.61
0.874
4.84
-10.07
0.60
3.10
0.0124
6.49
-0.1501
12.61
O.832
4.84
-10.08
0.50
3.43
0.0137
6.50
-0.2953
12.61
0.686
4.84
-10.09
0.40
3-45
0.0244
6.5^
-0.6375
12.61
0.384
4.85
-10.15
0.30
3.48
0.1246
5.46
-1.186
2.82
0.360
4.88
-10.42
0.25
3.49
0.2195
5^7
-1.462
2.82
0.360
4.89
-10.52
1.00
5.94
0.00423
4.89
f - 1/0.05 «
-0.00487
* 20 cps
12.61
0.967
4.84
-10.05
0.90
4.44
0.00359
4.92
-0.00909
12.61
0.962
4.84
-10.05
0.70
4.44
0.00296
6.47
-0.1022
12.61
0.868
4.84
-10.05
TABLE A. II VALUES OF MAXIMUM AND MINIMUM DEFORMATIONS
AND DISPLACEMENTS WITH THE ASSOCIATED TIMES
Elasto-Plastic Systems, Damping Factor, £ = 0,02, El Centro Earthquake
yQ = 8.26 in., yQ = 13.68 in. /sec., y = O.32 g, Duration of Quake = 29*5 sec.
u Deformations Absolute Displacements
uo
t
max
see.*
u
max
in.
t .
min
sec.
u ^
min
in.
t
max
sec .
x
max
in.
^min
sec .
x ^
min
in.
f * l/25 *
0.04 cps
1.00
10.92
11.03
25.39
-8.797
11.33
7.019
24.85
-6.695
0.80
10.92
11.03
4.33
-7.670
11.52
7.070
25-39
-2.662
0.70
11.00
11.05
4.33
-7.670
11.82
7.141
25.80
-0.739
0.60
10.92
9.909
4.33
-7.671
11.40
5.886
0
0
0.50
10.92
8.680
4.33
-7.677
10.92
4.6l6
0
0
o.4o
10.86
7.502
4-33
-7.686
10.48
3.463
0
0
0.30
22.00
7.028
4.33
-7.619
22.20
4.253
0
0
0.25
10.86
6.169
4-33
-7.686
21.60
2.951
0
0
0.20
10.86
5.589
4.33
-7.773
20.78
1.753
29-48
-O.038
0.15
10.86
5.287
4.33
-7.826
9.72
1.327
29.48
-0.591
0.10
10.86
4.866
4.33
-7.904
9-40
0.919
29.48
-O.382
0.05
10.86
4.585
M3
-7.992
9.26
O.606
29.48
-0.659
f « 1/15 -
O.O67 cps
1.00
10.80
12.87
15.58
-9.608
9.62
10.17
16.54
-9-224
0.80
10.80
12.94
4.33
-7.202
9.62
10.17
16.12
-4.540
0.70
10.80
13.10
M3
-7.202
9.72
10.17
15.90
-1.925
0.60
10.86*
13.47
4. 33
-7.202
9.80
10.24
0
0
0.50
10.86
12.45
M3
-7.205
22.00
9*4l4
0
0
0.40
10.86
10.32
M3
-7.222
21.59
7-465
0
0
0.30
10.86
8.795
M3
-7.040
9.40
5-551
0
0
0.25
10.86
8.161
4.33
-6.893
9.18
4.949
29.48
-1.251
0.20
10.86
6.832
4.33
-7.105
8.82
3.664
29.48
-3.254
0.15
10.86
6.936
4.33
-7.276
9.40
3.311
29-48
-2.489
0.10
10.86
5.928
M3
-7.478
9-04
2.275
29.48
-1.213
0.05
10.86
5.106
M3
-7.748
8.69
1.334
29.48
-I.658
f - 1/10 -
0.10 cps
1.00
28.23
17.09
13.29
-15.57
18.34
14.17
23.60
-15.34
0.90
28.23
16.81
13.29
-15.57
18.34
13.80
23.65
-15.37
0.80
28.23
13.35
13-78
-15.78
8.49
H.98
23.75
-15.76
0.70
8.80
10.52
13.78
-16.18
8.49
11.98
23.87
-16.36
0.60
8.80
10.52
24.80
-17.19
8.49
11.98
24.00
-16.68
0.50
8.80
10.59
24.85
-14.73
8.56
12.00
24.17
-13.69
0.40
8.82
10.18
24.89
-12.81
8.60
11.52
24.40
-11.26
A-77
TABLE A- 11 CONTINUED
u Deformations Absolute Displacements
u
0
Snax
sec.
umax
In.
t .
min
sec .
u .
min
In.
t
max
sec .
X
max
in.
^min
sec .
xmin
in.
0.30
IO.56
7.588
24.93
-12.51
8.49
8.474
24.40
-10.764
0.25
10.56
6.768
24.93
-12.01
8.33
7.581
24.48
-10.170
0.20
IO.56
6.080
24.93
-H.51
8.17
6.782
24.62
-9.568
0.15
10.80
6.092
24.93
-9-548
8.17
5.672
29.48
-7.791
0.10
10.86
7.687
4.33
-6.505
8.69
5.19^
29.48
-2.428
0.07
10.86
6.562
4.33
-6.884
8.56
3.767
29.48
-1.648
0.05
10.86
5.932
4.33
-7.176
8.44
2.871
29.48
-2.339
II
-4
II
0.143 cps
1.00
22.06
15.78
I8.69
-15.82
15.40
15.78
18.85
-15.20
0.90
15-41
14.53
25.40
-15.38
15-40
15.78
18.95
-14.71
0.80
15.74
14.04
11.82
-13,03
15-41
15.13
11.80
-13.83
0.70
15.74
H.16
11.82
-13.04
7.68
13.68
11.80
-13.84
0.60
8.33
9.706
11.82
-12.68
7.68
13.68
11.82
-.13.48
0.50
6.33
8.907
11.82
-10.41
7.68
12.75
11.82
-11.20
0.40
2.60
6.211
12.60
-10.95
7.70
9.544
12.08
-11.02
0.30
2.60
6.240
12.92
-10.41
7.60
8.042
12.20
-9.753
0.25
2.63
6.281
13.00
-10.12
7.40
7.474
12.33
-9.026
0.20
2.67
6-399
13.00
-9.279
7.33
7.423
29-48
-8.561
0.15
2.67
6.268
24.93
-7.833
7.70
8.212
29.48
-6.637
0.10
10.80
7.183
24.93
-6.342
8.17
7.321
24.24
-4.625
0.07
10.80
6.409
24.93
-6.4n
8.17
5.533
24.07
-4.732
0.05
10.80
5.983
^•33
-6.519
8.17
4.322
24.00
-4.639
0.03
10.86
5.724
4.33
-7-042
8.17
3.106
24.00
-4.243
f - 1/5
» 0.2 cps
1.00
26.40
10.44
28.56
-12.02
6.38
15.52
28.68
-15.OO
0.80
26.40
8.172
28.68
-12.89
6.38
12.71
28.68
-15.89
0.70
2.55
6.475
28.68
-12.59
6.38
1Q.20
28.68
-15.60
0.60
2.55
6.475
28.68
-12.31
6.48
7-5^1
28.68
-15.32
0.50
2.55
6.481
28.68
-11.27
6.53
5.488
28.68
-14.28
0.40
2-55
6.583
4.21
-9.219
6.36
5.014
22.80
-11.19
0.30
26.53
8.602
4.21
-6.462
26.21
6.997
H.16
-6.963
0.25
26.64
10.03
4.20
-4.970
7.25
8.598
23.19
-4.629
0.22
26.64
9.257
4.20
-4.050
7.33
9.893
23.39
-4.003
0.20
27.44
7.887
4.20
-3.700
7.40
10.44
23.52
-4.296
0.15
8.44
7.631
24.78
-7.063
7.70
10.48
25.75
-6.998
0.10
2.64
6.271
24.85
-9.770
7.68
8.287
23.75
-9.136
0.05
10.80
5.^34
24.89
-8.839
7.79
5.297
23.65
-7.629
TABLE A.li CONTINUED
u
_2
Deformations
Absolute Displacements
u
0
t
max
sec .
u
max
in.
^min
sec .
Umin
in.
t
max
sec .
X
max
in.
^min
sec .
xmin
in.
1.00
5.40
13.91
3.94
f = l/4 :
-II.85
=0.25 cps
5-52
18.51
27.96
-13-58
0.80
5.40
12.73
3.96
-11.87
5.61
17.49
28.08
-11.18
0.70
5-34
9-845
3.96
-12.06
5.61
14.51
28.08
-12.11
0.60
5.34
6.302
4.01
-12.57
5.61
10.86
28.11
-13. 71
0.50
2.33
6.050
4.01
-13.42
5.61
6.590
28.20
-15.99
0.40
2.33
6.067
4.08
-13.46
1.95
4.269
28.23
-15.87
0.50
2.55
6.581
4.08
-IO.06
5.52
4.389
28.32
-9.833
0.25
2-55
7.122
4.08
-7.874
7.32
6.233
28.35
-6.216
0.20
2.55
7.744
4.08
-5.537
7.40
9.137
11.76
-3.931
0.15
26.53
10.82
4.08
-3.470
7-40
11.87
0
0
0.10
26.64
8.427
4.17
-3.706
7-40
10.48
22.88
-2.526
0.05
2.64
5-945
24.85
-9.220
7.46
7.211
23.64
-8.724
1.00
13.62
18. 04
9.26
f - 1/3 -
-17.72
O.33 cps
7.70
21.97
12.12
-18.39
0.90
4.98
15.66
9.30
-17.82
4.86
21.49
12.12
-18.64
0.80
5-04
15.82
9-30
-15.40
4.86
21.53
12.12
-16.44
0.70
5.04
16.49
3.36
-H.56
4.92
21.78
12.08
-12.24
0.60
507
15.75
306
-H.58
7-70
21.30
3.24
-9.702
0.50
5.10
12.62
3-37
-11.76
7.70
18.07
3.24
-9.733
0.40
5.16
9.435
3.42
-12.21
7.68
14.31
3.32
-9.953
0.30
5.22
6.375
3.84
-11.82
7.56
11. 14
2700
-8.813
0.25
1.90
6.223
3.90
-10.77
7.46
10.35
27.36
-9.486
0.20
1.90
6.336
4.01
-9.915
7.38
9.434
27.42
-10.75
0.15
2.33
6.751
4.02
-8.609
7.26
9.141
27.58
-10.65
0.10
2.55
8.222
12.48
-5.208
7.14
11.05
27.84*
-8.104
0.05
2.58
7.223
12.92
-5.119
7.09
9.503
22.94
-5.772
1.00
7.01
15.89
5.80
f = 1/2.5
-15.80
* 0.4 cps
7.01
23.29
10.92
-15.76
0.80
4.57
12.09
5.82
-16.13
4.50
18.96
10.98
-16.96
0.70
4.57
12.12
5.82
-14.36
4.50
18.96
11.00
-15.79
0.60
4.62
11.82
5.88
-11.55
4.50
18.51
11.05
-13.72
0.50
4.62
8.891
5.88
-11.33
4.56
15.31
11.10
-14.42
0.40
4.68
5.936
11.22
-11.26
4.62
11.99
11.22
-15.19
0.30
1.86
5.823
11.70
-10.47
4.76
IO.83
11.40-
-13.55
0.25
5.10
5.986
3.36
-7.918
4.86
11.03
27.30
-9.981
0.20
5.16
3-37
-6.623
7.09
13.12
27.42
-7.298
0.15
10.20
3.90
-6.292
7.02
13.03
27.54
-7.774
0.10
2*33
7.247
27.36
-6.262
6.96
9.443
27.66
-9.332
0.05
2.50
7.313
12.60
-6.848
6.91
9.279
22.80
-7.554
A-79
TABLE A. 11 CONTINUED
u Deformations Absolute Displacements
uo
t
max
umax
min
Slin
t
max
xmax
^min
x .
min
sec.
in.
sec.
in.
sec.
in.
sec.
in.
f « l/2 -
O.5O cps
1.00
12.33
7.486
11.34
-6.710
6.47
13.75
11.33
10.53
0.90
12.33
7.531
11.34
-6.710
6.47
13.75
11.33
-10.53
0.80
12.36
6.049
11.40
-6.779
6.47
13-75
11.33
-10.57
0.70
6.53
5.952
11.40
-5.819
6.48
14.00
11.34
-9.547
0.60
6,60
6.190
2.94
-5.127
6.56
14.12
2.70
-8.272
0.50
6.60
4.554
5-70
-5.142
6.60
12.41
11.34
-7.743
0.40
1.80
4.636
11.64
-7.045
4.26
10.62
11.34
-10.45
0.50
1.80
5-014
11.70
-6.299
4.50
IO.50
11.33
-9.489
0.25
1.80
5-357
11.70
-5.610
4.68
IO.83
11.28
-8.705
0.20
5.16
7.622
3*37
-2.674
6.78
12.49
22.20
-3-481
0.15
1,90
6.753
4.01
-4.359
6.72
11.05
27.36
-6.425
0.10
2.55
7.362
11.82
-6.575
6.72
9.196
27.77
-8.896
0.05
2.58
6.591
12.96
-8.253
6.84
8.411
22.80
-9.770
f - 1/1.50
■ O.67 cps
1.00
6.30
5.166
8.61
-4.998
6.27
13.38
11.55
-7.136
0.80
12.27
5.390
5.61
-3.489
6.30
13.50
2.33
-5-2T7
0.70
9.48
5.531
5.61
-3.489
6.33
13.69
2.33
-5.277
0.60
6.36
4.556
5.61
-3.735
6.36
12.73
2.34
-5.333
0.50
27.58
2.359
5.67
-4.741
6.39
10.44
28.29
-6.076
0.40
1.74
1.868
5.76
-6.030
6.42
7.670
28.26
-8.671
0.30
1.74
2.223
28.32
-5.451
4.50
8.044
28.29
-8.466
0.25
1.77
2.893
28.32
-5.316
4.50
8.273
28.29
-8.327
0.20
1.80
3.827
27.03
-5.436
4.53
8.544
28,29
-8.407
0.15
5.15
5-759
3.36
-1.815
4.83
IO.55
29.48
-4.597
0.10
2.55
5.813
4.01
-4.238
6.54
8.767
28.14
-6.469
0.05
2.58
6,654
12.90
-6.649
6.66
8.927
28.14
-8.088
f - 1/1.25
■ 0.80 cps
1.00
6.15
5.179
6.78
-4.6a
6.15
13.44
27.99
-5.527
0.80
6.18
5.309
5.52
-3.678
6.18
13.58
27.99
-4.424
0.70
15.84
5.516
5.55
-3.679
6,20
13.67
2901
-4.030
0.60
15.81
5-546
5.55
-3.273
6. a
13.U
2.22
-3.921
0.50
15.81
5.096
5.58
-3.442
6.24
12.02
2.22
-3.924
0.40
15.81
4.877
3.16
-3.458
6.27
11.08
2.94
-4.196
0.30
15.84
4.263
3.16
-3.832
6.32
9.637
11.16
-5.3^3
0.25
0.20
15. §4
15.84
m
5-73
5.79
-4.122
-4.801
6.36
4.56
m
11.16
11.22
-6.566
-7.995
0.15
1.77
2.183
8.67
-5.997
4.59
6.599
11.19
-9.291
0.10
1.83
3.590
3.37
-4.154
4.86
8.165
11.16
-7.220
0.05
2.55
6.479
11.79
-4.302
6.51
8.948
29.48
-7.291
TABLE A. 11 CONTINUED
u Deformations Absolute Displacements
u
0
t
max
sec .
u
max
in.
"^min
sec .
Umin
in.
t
max
sec .
X
max
in.
^min
sec .
Xmin
in.
1.00
4.50
5.967
4.95
f = 1
-5.968
cps
4.47
13.168
IO.89
-6.797
0.80
4.52
6.114
3-99
-4.640
4.47
13.21
2.94
-5.364
0.70
4.53
5.934
3-99
-3*994
4.47
12.98
■ 2.94
-5.364
0.60
4.53
5-795
3.03
-3.735
4.47
12.78
2.94
-5.364
0.50
4.56
4.638
3.06
-3.835
4.50-
11.57
2.94
-5.366
0.4o
12.14
3.991
3.06
-4.095
6.15
10.94
2.94
-5.504
0.50
8.13
2.584
3.O8
-4.328
6.12.
10.78
2.91
-5.749
0.25
8.17
2.913
3.08
-4.451
6.15
11.01
2.91
-5.897
0.20
8.19
3.OO7
3-12
-4.498
6.20
11.03
2.88
-5.942
0.15
2703
2.173
3.16
-4.499
6.24
9.488
2.87
-5.730
0.10
1.80
1.505
5-79
-5.089
4.57
6.104
11.04
-8.668
0.05
I.83
2.746
H.76
-5.937
4.86
6.244
20.19
-9.091
1.00
5.87
3-320
5.48
f = 1/0.7 «
-3.144
= 1.43 cps
5-87
11.35
2.72
-6.754
0.80
2.34
2.703
5.48
-3-167
5.88
10.48
2.72
-6.670
0.70
2.34
2.727
5.49
-2.670
5.89
10.41
2.72
-6.036
0.60
2.34
2.181
5.49
-2.768
5.91
9-755
2.72
-5.944
0.50
2.34
1.403
5.49
-3.138
5-93
8.869
11.12
-6.122
0.40
1.74
I.013
5.51
-3.761
5.96
7-778
11.12
-6.585
0.30
1.74
1.013
5.58
-4.219
6.00
7.025
2.75
-6.886
0.25
1.74
1.024
5.61
-4.584
4.25
6.541
11.10
-7.273
0.20
1.76
1.116
5.64
-4.971
4.31
6.929
11.09
-7.754
0.15
1.76
1.316
5.69
-4.052
4.38
7.199
11.04
-6.765
0.10
1.77
1.766
3.18
-3.492
4.47
7.267
11.01
-6.301
0.05
1.82
2.620
11.76
-4.913
4.67
5-904
11.01
-7.750
1.00
3.17
1.853
2.43
f * 1/O.5
-2.242
■ 2 cps
6.05
9.675
2.45
-5.297
0.80
2.21
1.833
5.30
-2.540
6.03
8.686
2.46
-5.309
0.70
2.21
1.865
5.31
-2.142
6.03
8.673
27.72
-5.080
0.60
2.22
1.942
26.70
-1.412
6.03
9.028
27.72
-4.542
0.50
5.04
1.811
1.91
-1,332
6.0?
8.949
27.74
-4.519
0.40
2.27
1.450
1.94
-1.471
4.16
8.526
27-75
-4.526
0.30
5.07
1-359
2.01
-I.58O
4.44
8.549
27.75
-4.154
0.25
5.07
I.I63
2.04
-i.712
4.46
8.249
11.00
-4.497
0.20
1.68
1.233
2.07
-1.596
6.36
7.9^8
11.03
-4.923
0.15
1.74
1.696
5.58
-1.372
5.99
8.208
11.05
-4.677
0.10
1.77
1.882
5.66
-2.339
4.33
7.982
10.95
-5.702
0.05
1.80
2.570
3.90
-3.517
4.53
6.521
10.94
-6.243
TABLE A.il CONTINUED
uv Deformations Absolute Displacements
u
0
t
max
sec.
u
max
in.
min
sec .
u .
min
in.
t
max
sec .
xmax
in.
min
sec .
Xmin
in.
1.00
4.95
1.323
5.17
f - 1/0.4 =
-1.470
»2.5 cps
6,18
9-153
2.76
-5.028
0.80
2.57
1.104
5.18
-1.607
6.20
8.559
2.76
-5.025
0.70
2.57
0.907
5.18
-1.619
6.21
8.313
2.76
-5.049
o.6o
2.19
0.708
5.18
-1.641
6.21
8.069
10.92
-5.220
0.50
2,19
0.708
26.09
-1.831
6.23
7.853
10.92
-5.378
0.40
2.21
0.769
5-21
-1.482
6.24
7-855
10.94
-4.996
0.30
2.24
1.219
5.27
-0.881
6.27
8.298
10.95
-4.397
0.25
2.28
1.282
1.98
-0.647
6.30
8.549
10.97
-4.154
0.20
I.65
O.962
26.66
-1.223
4.37
7.9U
10.98
-4.842
0.15
1.71
1.282
26.72
-1.962
4.38
7.266
11.00
-5.581
0.10
1.76
1-572
5.64
-2.443
6.05
6.906
10.98
-5.822
0.05
1.80
2.385
3.89
-3-184
4.49
6.481
10.97
-6.250
1.00
2.50
0.520
2.64
f - 1/O.3 -
-0.752
3.33 cps
6.19
8.465
2.64
-4.687
0.80
2.50
0.520
2.65
-0.769
6.20
8.327
2.65
-4.711
0.70
2.50
0.515
2.65
-0.818
6.20
8.225
2.66
-4.764
0.60
2.50
0.579
10.14
-0.686
'6.21
8.296
2.66
-4.550
0.50
2.52
0.700
9.53
-0.419
4.37
8.499
2.67
-4.245
0.40
9.36
0.692
1-33
-0.268
6.22
8.638
2.68
-4.126
0.30
25.78
0.936
1.34
-0.273
6.21
8.868
2.69
-4.040
0.25
2.25
1.147
1.34
-0.274
6.23
8.948
2.70
-3.803
0.22
2.27
1.061
2.00
-0.357
6.22
8.735
IO.90
-3.822
0.20
2.26
0.751
2.03
-0.659
6.16
8.314
10.91
-4.245
0.15
1.70
0.919
3.57.
-1.433
6.22
7-134
10.94
-5.274
0.10
1.76
1.453
5.63
-2.443
6.28
6.534
10.96
-5.970
0.05
1.80
2.373
5.78
-3.048
4.46
6.347
11.00
-6.388
1.00
2.96
0.667
2.60
f - 1/0.25
-0.746
- 4 cps
6.20
8.426
2.60
-4.650
0.80
2.46
O.582
2.61
-0.761
6.19
8.241
2.61
-4.672
0.70
2.47
O.588
2.61
-0.656
6.18
8.259
2.61
-4.567
0.60
2.47
0.507
2.62
-0.615
6.18
8.211
2*62
-4.531
0.50
2.49
0.457
2.62
-0.539
6.18
8.199
2.63
-4.462
0.40
2.50
0.308
4.77
-0.567
4.36
8.111
2.64
-4.487
0.30
22.47
0.404
2.35
-O.568
6.21
8.211
2.66
-4.505
0.25
24.55
0.564
3.54
-0.537
6.22
8.242
2.66
-4.437
0,20
24.57
0.549
1.96
-0.422
6.22
8.291
2.67
-4.300
0.15
26.19
0.837
2.00
-0.361
6.14
8.655
2.69
-3.924
0.10
1.70
0.716
3.56
-1.400
6.22
7.338
10.91
-5-017
0.05
1.78
1.586
5.70
-2.788
6.14
6.578
10.98
-5.870
TABLE A -11 CONTINUED
u
JL
Deformations
Absolute Displacements
u
0
t
max
sec .
u
max
in.
Snin
sec .
Umin
in.
t
max
sec.
X
max
in.
^min
sec .
Xmin
in.
1.00
3.30
0.415
2.78
f = 1/0.20 =
-0.423
* 5 cps
6.24
8.482
2-59
-4.243
0.80
2.69
0.279
2.79
-0.435
6. 24
8.412
2.59
-4.244
0.70
2.48
0.227
5.13
-0.448
6.24
8.311
10.90
-4.325
0.60
2.48
0.227
3.01
-0.472
6.24
8.235
10.90
-4.381
0.50
2.48
0.228
3.01
-0.479
6.24
8.215
10.90
-4.380
0.40
2.48
0.246
26.06
-0 . 423
6.25
8.166
10.90
-4.406
0.30
2.49
0.350
5-14
-0.396
4.31
8.133
10.90
-4.232
0.25
2.31
0.476
5.14
-0 . 271
4.30
8.283
10.92
-4.080
0.20
2.22
0.278
5-15
-0.464
4.32
8.089
10.94
-4.214
0.15
2.26
O.303
2.02
-0.701
6.23
8.099
2.68
-4.296
0.10
1.70
0.541
2.92
-1.490
6.22
7.469
IO.89
-4.8i4
1.00
4.95
0.262
5.03
f = 1/0.15 =
-O.213
6.67 cps
4.33
8.356
IO.89
-4.157
0.80
4.96
0.286
3.65
-0.212
6.13
8.406
IO.89
-4.08l
0.70
4.96
0.275
3.65
-0.214
6.13
8.407
10.89
-4.065
0.60
4.97
0.280
3.65
-0.202
6.13
8.425
10.89
-4.035
0.50
4.98
0.314
3-66
-0.181
6.24
8.480
2.70
-4.012
0.40
5.81
O.388
3*67
-0.136
6.14
8.588
2.71
-3.987
0.30
5-00
0.309
• 3.68
-0.241
6.15
8.489
10.90
-4.003
0.25
5.01
0.315
3.68
-0.248
6.1 6
8.485
IO.90
-3-979
0.20
2.30
0.374
3.69
-0.133
6.17
8.523
IO.90
-3.902
0.10
26.37
0.236
3-73
-0.911
6.20
8.002
2.70
-4.287
1.000
4.95
0.114
5.38
f « 1/0.125
-0.106
m 8 CpS
6.20
8.302
10.86
-4.097
0.716
4.95
0.121
2.73
-0.089
6.20
8.330
10.86
-4.059
0.626
4.96
0.l4l
2.73
-0.079
6.21
8.357
10.86
-4.029
0.537
4.97
0.172
2.35
-0.074
6.21
8.396
10.86
-3.986
0.447
13.83
0.207
2.36
-0.066
6.22
8.425
2.70
-3.958
0.358
2.49
O.169
2.78
-0.152
6.22
8.297
10.86
-4.043
0.268
5.02
0.416
3-19
-0.092
6.23
8.546
2.65
-3-912
0.224
26.30
0.380
1.99
-0.233
6.15
8.516
2.66
-4.043
0.179
26.34
O.634
2.03
-0.535
6.16
8.447
2.68
-4.119
0.134
1.70
0.315
3-73
-1.350
6.20
7.610
10.90
-4. 616
1.00
4.79
0.0575
2.69
f * 1/0.10 .
-0.0495
■ 10 cps
6.20
8.292
10.90
-4.085
0.80
4.79
0.0561
2.69
-0.0498
6.20
8.301
10.90
-4.075
0.70
9.58
0.0578
0.0899
2.70
-0.0504
6.20
8.304
10.90
-4.069
0.60
26.30
2.70
-0.0421
6.20
8.328
10.90
-4.0J6
-3.987
0.50
26.30
0^1417
4.74
-0.0425
6.20
1:212
IO.90
o.m
26.30
0.2096
4.74
-0.0714
6.20
2.70
-3-926
TABLE A. 11 CONTINUED
u Deformations Absolute Displacements
•JL ■ ■ — IP ■ — — . ■■■Lilli ,1—
u
0
t
max
sec.
u
max
in.
t .
min
sec .
u .
min
in.
t
max
sec .
X
max
in.
t
min
sec .
Xmin
in.
0.55
26.50
0.1797
2.81
-0.155
6.20
8.564
2.70
-5.961
0.50
5.02
O.II85
5.02
-0.2709
6.19
8.206
IO.89
-4.089
0.25
26.29
0,5559
2.01
-O.5496
f = 1/0.05
6.19
= 20 cps
8.544
2.66
-4.045
1.00
9-55
0.01009
2.56
-O.OO856
6.19
8.278
10.89
-4.069
0.90
9.56
O.OO935
2.56
-0.00925
6.19
8.278
10.88
-4.069
0.80
9.56
0.0115
2.5l
-0.0075
6.19
8.280
10.88
-4.066
0.50
11.27
0.1566
1.9l
-0.0528
6.19
8.596
10.88
-5.959
TABLE A. 12a
MAXIMUM DEFORMATIONS OF SINGLE DEGREE -OF -FREEDOM BILINEAR SYSTEMS
2 Percent Critical Damping; Eureka Earthquake
*2/*!
- 0.75
- 0.50
Vuo
- 0.25
tmax
u
max
^max
u
max
t
max
Umax
t
max
u
max
1.0
18.06
- 9.62
18.06
t - 0.10
- 9.62
cps
18.06
- 9.62
18.06
- 9.62
0.9
18.16
- 9.51
18.26
- 9.25
18.5*
- 9.55
18.5*
- 9.98
0.8
18.20
- 8.99
15.58
8.82
15.66
9.58
15.66
10.00
0.7
7.**
- 8.87
15.60
8.55
15.78
8.99
15.92
9-79
0.6
7.**
- 8.85
15.66
8.05
15.98
8.18
1*.58
9.05
0.5
7.4*
- 8.79
*.60
7.89
*.70
8.2*
*.80
8.5*
0.*
7.**
- 8.7*
*.70
7.95
*.80
8.58
*.80
8.76
0.5
7.**
- 8.7O
*•70
7.98
*.80
8.55
*.80
8.97
0.2
7.**
- 8.65
*•70
8.05
*.80
8.68
*.80
9.19
0.1
7.**
- 8.59
*.70
8.08
*.80
8.85
*.80
9.*2
0
7.**
- 8.55
*.70
8.15
*.80
8.98
*.80
9.6*
1.0
8.28
-10.75
8.28
r - 1/5
-10.75
cps
8.28
-10.75
8.28
-10.75
0.9
6.78
IO.58
6.8*
10.5*
6.8*
10. *2
1*.88
-10.92
0.8
6.78
10.61
6.8*
10.12
6.92
10.16
7.02
10.99
0.7
6.78
10.6*
6.88
9.82
7.06
9.77
7.20
11.06
0.6
6.78
10.67
6.90
9. *7
7.1*
9.21
7.50
10.70
0.5
6.8*
10.72
6.92
9.05
5.56
- 8.59
5.6*
-10.05
0.*
6.8*
10.76
6.96
8.5*
5.58
- 8.52
5.76
-10.10
0.5
6.8*
10.81
5**6
- 8.56
5.60
- 8.55
5.82
- 9.7*
0.2
6.8*
10.86
5**6
- 8. *6
5.6*
- 8.08
6.12
- 8.65
0.1
6.8*
10.91
5.*6
- 8.57
5.70
- 7.71
6.18
- 6.88
0
6.8*
10.96
5.52
- 8.68
5.76
- 7.21
5.90
5.79
1.0
5.07
-5.506
0.9
5.07
-5.521
0.8
5.07
-5.525
0.7
5.07
-5.551
0.6
5.07
-5.566
0.5
5.07
-5.581
0.*
5.07
-3 *397
0.5
5.07
-5**13
0.2
5.07
-5. *28
0.1
5.08
-3»**6
0
5.08
-5 .*65
1.0
6.01
-0.151
0.9
6.01
-0.1*5
0.8
6.01
-0.1*0
0.7
6.01
-0.157
0.6
*.82
-0.155
0.5
*.85
-0.156
0.*
*.85
-0.156
0.5
0.2
*.85
*.85
-0.157
-0.138
0.1
6.01
-0.1*0
0.075
•
•
0.05
-
-
0.025
-
-
0.01
•
•
0
6.01
-0.1*7
f ■ 1 cps
5-07
-3.306
5.07
-3.306
5.07
-3.337
5.08
-3- *35
5.07
-3.362
5.10
-3 .*91
5.08
-3.382
5.13
-3- *81
5.08
-3* *05
5.16
-3. *00
5.08
-3. *23
5.19
-3.278
5.10
5.10
5.20
5.22
-3.130
-3.029
5.10
-3. *82
5.25
-3.039
5.13
5.13
m
l:U
3:1%
6.01
f - 5 cps
-0.151
6.01
-0.151
*.§3
-O.132
*.8*
-0.159
-0.1*3
*.8*
-0.129
*.86
*.8*
-0.129
*.87
-0.1*9
*.85
-0.129
*.89
-0.160
*.85
-O.131
*.90
-0.179
*.86
-O.136
*•93
-0.212
*.86
-0.1*6
*.9*
-0.2*9
*.86
-O.163
*.97
5.00
-0.300
*.87
-O.19*
-0.*30
*.89
-0.206
5.01
-0.502
6.00
-0.222
5.02
-0.620
6.01
-0.262
5.05
-0.8*1
6^02
-07313
A-85
5.07 -5.J06
5.08 .5.709
5.15 -5.981
5.16 -*.069
5.9* *.011
6.00
*.18
*.26
*.52
5.21
5.61
5.757
-2.876
-5.O6O
-2.950
-*.12
-*.3
6.01
6.18
*•97
6.27
f£
5.0?
5.06
5.12
m
-0.151
0.150
-0.165
-0.199
-0.2*5
0.529
-o.;
0.1
0.*86
-0.5**
TABLE A. 12b
MAXIMUM DEFORMATIONS OF SIIC&E -DEGREE -QF-FRBEDOM BILINEAR SYSTEMS
2 Percent Critical Duping; El Centro Earthquake
Vu«
» 0.75
Y“o
- 0.50
- 0.25
Vuo
■ 0.10
t
■ax
u
■ax
*max
u
max
t
max
u y
t
umax
1.0
28.23
17.09
t
28.23
■ 0.10 cpa
17.09
28.23
17.09
28.23
17.09
0.9
28.23
16. k5
13.78
-15. kO
13.78
-lk.79
2k. 80
-15.33
0.8
28.23
16.03
13.78
-15.25
lk.26
-lk.06
2k. 93
-lk.59
0.7
28.23
15.78
13.78
-15.06
lk.26
-13. k9
lk.26
-13.51
0.6
13.78
-15.69
13.78
-lk.8k
lk.26
-12.8k
lk.26
-12.0k
0.5
13.78
-15.7k
lk. 26
-lk.79
lk.26
-12. k 5
10.56
10.19
o.k
13.78
-15.78
lk. 26
-lk.55
lk.26
-11.29
10.80
10.32
0.3
13.78
-15.83
lk.26
-lk.35
2k. 93
-H.06
10.80
10.27
0.2
13.78
-15.87
2k. 80
-15.00
2k. 89
-12.10
10.80
9.82
0.1
13.78
-15.92
2k. 85
-lk.90
2k. 93
-11.9k
10.86
8.98
0
13-78
-15.96
2k. 85
-lk.73
2k. 93
-12.01
10.86
7.69
1.0
13.62
18.0k
13.62
f - 1/3 cpa
18.0k
13.62
18.0k
13.62
18.0k
0.9
9.30
-16.69
5.0k
15.58
5.0k
15.65
5.06
16.05
0.8
6.72
-15-93
5.0k
15.50
5.10
15 .k5
5.15
16.00
0.7
k.98
15.77
5.06
15.35
5.15
15.05
5.22
15.5k
0.6
k.98
15.80
5.06
15.16
5.16
lk.k7
5.28
lk.k3
0.5
5.0k
15.85
5.06
lk.91
5.22
13-73
5.3k
12.55
O.k
5.0k
15.90
5.10
lk.6l
5.22
12.70
3.96
-11.12
0.3
5.0k
15.96
5.10
lk.22
5.22
11. k2
k.01
-11.11
0.2
5.0k
16.01
5.10
13.75
3.78
-10.66
k.02
-10.16
0.1
5-Ok
16.06
5.10
13.22
3.8k
-10.7k
k.08
- 8*29
0
5.0k
16.12
5.10
12.6 2
3.90
-10.77
2.55
8.22
1.0
k.95
-5.97
k.95
f - 1 cpa
-5-97
k.95
-5.97
k.95
-5.97
0.9
k.50
5.95
k.52
5.59
k.53
5.30
k.56
5.32
0.8
k.50
5.9k
k.53
5.22
k.57
k.57
k.59
k.37
0.7
k.52
5.92
k.53
k.92
k.59
3.92
k.62
3-k5
0.6
k.52
5.92
k.53
k.69
k.59
3.50
6.11
3.07
0.5
k.52
5.91
k.53
k.59
k.59
3.30
6.18
3.09
O.k
k.52
5.91
k.53
k.60
3.06
-3.39
6.2k
2.99
o.3
k.52
5.90
k.53
k.66
3.06
-3.70
3.08
-3.15
0.2
k.52
5.90
k.53
k.72
3.06
-k.17
3.09
-3.77
0.1
k.52
5.89
k.53
k.73
3.08
-k.30
3*15
-k.10
0
k.52
5.89
k.56
k.6k
3.09
-k.k5
5.79
-5.09
1.0
2.78
-0.k23
2.78
f - 5 cpa
-0.k23
2.78
-0.k23
2.78
-0.k23
0.9
2.78
-0.k23
2.79
-0.399
2.79
-0.395
2.80
-O.ki.5
0.8
2.78
-0.k25
2.79
-0.378
2.80
-0.362
2.81
-0.376
0.7
2.78
-0.k27
2.79
-O.36O
2.60
-0.355
2.60
-0.393
0.6
2.78
-0.k28
2.59
-0.357
2.61
-0.359
2.61
-0.512
0.5
2.78
-0.k29
2.59
-0.358
2.61
— 0.368
2.63
-0.6k8
O.k
2.78
-0.k30
2.79
-O.363
2.62
-0.375
2.66
-O.673
0.3
2.79
-0.k33
2.79
-0.39k
2.63
-0.362
2.5k
0.6kl
0.2
2.79
-0.k35
2.80
-o.kko
2.82
-0.328
2.k2
-O.867
0.1
2.79
-0.k38
2.80
-0.k55
2.50
0.35k
2.26
I.089
0
2.79
-O.kkl
3.01
-0.k79
2.51
0.k76
2.k2
-l.k90
APPKSDIX B
EXPRESSIONS TOR RBSPOHSB OF A SIMHJg-DEraEE-Cg -FREEDOM ST8TB4
The solution of Eq. 2.k presented in the body of this report may
be expressed by means of Duhamel's Integral as
u(t) -
e“ft?t £u(0) cos pdt + + 3 u(0)j sin pdtj
L Jl7 ' ' J
t T
— f Kr) .in [p4(t-r)l dt
£7
(B.l)
vhere t Is tine, t Is a variable of Integration, and pd, the circular natural
frequency of the danped system, is related to the corresponding undated
frequency by the equation
Pd - p ^1-? (B.2)
The acceleration function f(r) Is considered to be sectlonally continuous In
the Interval between 0 and t.
The expression far the relative velocity, Is obtained by
differentiating Eq. B.l with respect to tine
u(t) - e"^1* |ju(0) cos pdt - (p^0) ♦ P «(°)) >ln Pd*]
t
- f y(t) fcos[pd(t-T)] - — sin[pd(t-T)]ldT (B.
0 L ^7
The absolute acceleration of the mass, 'i, is obtained by a further differentia*
tlon or, more conveniently, directly from Eq. (2.2) by substituting for u and &
the expressions given by Iqs. B.l and B.?. The resulting expression Is
3£(t) - [no)
t
♦ P f ¥(0
cos p^t -
£7
e-0p(t-T) [1^P2
l-2p2) pu(0) - 3 p2u(0)j sin pdt"|
sln[pd(t-T)] ♦ 23 eo«[pd(t-T)]ldT
(B.M
The regaining response quantities, 8, i and x, can now be date rained
froa the equations
U ■ x - y, u ■ x - f, and u ■ x - jr,
respectively. Vote that before x and x can be evaluated, the quantities y and
y aust be determined by integrating the prescribed acceleration function.
If the ground notion Is specified as a velocity-tine function, the
relative displacement, u, the absolute and relative velocities, i and u, and
the absolute acceleration, % can be expressed directly In terns of the
prescribed function as follows . The expression for u is obtained by inte¬
grating the last tern In lq. B.l by parts and c cabining terns.
u(t) • e"ft?t ju(0) cos pdt + + 3 u(0)J sin PdtJ
t
- J y(0 e"pp^t"T^ [cos[pd(t-T>] - Bin ^pd^t"T^]dT (B,5)
(B.6)
With x known, the relative velocity can be determined from £ ■ ft + $~. Finally,
2 can be determined from Xq. 2.2.
If the ground notion is specified as a displacement function, the
absolute displacement of the mass can be expressed in terms of the prescribed
function by integrating lq. B.5 *7 parts* The result is
B-2
x(t) . .■»* [x( 0) CO. p4t ♦ -A- (i^- ♦ »W0) - 2jr(0)l) .in P#t]
£7
* V f K») e"ap^t_T^ .ln[p4(t-l)] + 2p co.lp4(t-0)]dt (»*7)
'o L£7
Differentiation of this equation gives the absolute velocity,
i(t) - 2pp y(t)
♦ .-»* [ (i(0) - 2fSpy(0)l CO. p4t - -A- (pt*(0) - 2@2x(0)1 + f> *«>} •!« Pjt]
L &-? \
+ p2 frO) e*ep(t’t) [(lV)cos[p4(t-t)l - sin[pd(t-t)]]dt (B.8)
0 L £-?
»-5
RTD TDR-63-3096, Vol III
No, cvs
3
1
1
1
1
1
1
1
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2
2
2
1
1
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1
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2
2
1
1
1
DISTRIBUTION
HEADQUARTERS USAF
Hq USAF (AFOCE), Wash, DC 20330
Hq USAF (AFRDP), Wash, DC 20330
Hq USAF (AFRNE-A, Maj Griesmer), Wash, DC 20330
Hq USAF (AFTAC), Wash, DC 20330
USAF Dep, The Inspector General (AFIDI), Norton AFB, Calif
92409
USAF Directorate of Nuclear Safety (AF INS), KirtlandAFB, NM
87117
MAJOR AIR COMMANDS
AFSC, Andrews AFB, Wash, DC 20331
( SCT)
(SCLT)
(SCMC)
TAC, ATTN: Director of Civil Engineering, Langley AFB, Va 23365
SAC, ATTN: Director of Civil Engineering, Offut AFB, Nebr 57113
ADC, ATTN: Director of Civil Engineering, Ent AFB, Colorado
Springs, Colo 80912
AUL, Maxwell AFB, Ala 36112
USAFIT, Wright-Patter son AFB, Ohio 45433
USAFE, ATTN: Director of Civil Engineering, APO 633, New
York, NY
PACAF, ATTN: Director of Civil Engineering, Camp Smith, Hawaii
AFSC ORGANIZATIONS
AFSC Scientific and Technical Liaison Office, Research and
Technology Division (AFUPO), Los Angeles, Calif 90045
ASD, Wright-Patter son AFB, Ohio 45433
(SEPIR)
(ASAMC)
RTD, Bolling AFB, Wash, DC 20332
( RTN)
(RTN-W, Lt Col Munyon)
( RTS)
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RTD TDR-63-3096, Vol III
No. cvs
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1
2
2
2
3
1
1
1
1
1
2
1
1
1
1
20
2
10
1
1
1
1
1
1
DISTRIBUTION (cont'd)
BSD, Norton AFB, Calif 92409
( BSR)
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( BSSF)
(Document Library)
SSD(SSN), AF Unit Post Office, Los Angeles, Calif 90045
ESD, L. G. Hanscom Fid, Bedford, Mass 01731
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KIRTLAND AFB ORGANIZATIONS
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( SWEH)
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AFWL, Kirtland AFB, NM 87117
(WLIL)
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ADC (ADSWO), Special Weapons Office, Kirtland AFB, NM 87117
SAC Res Rep(SACLO), AFSWC, Kirtland AFB, NM 87117
TAC Liaison Office (TACLO-S), AFSWC, Kirtland AFB, NM 87117
OTHER AIR FORCE AGENCIES
USAF Engineering Liaison Office, APO 125, New York, NY
AFOAR, Bldg T-D, Wash, DC 20333
AFOSR, Bldg T-D, Wash, DC 20333
C-2
RTD TDR-63-3096, Vol III
t
DISTRIBUTION (cont'd)
No, cvs
2 Director, USAF Project RAND, via: Air Force Liaison Office,
The RAND Corporation, 1700 Main Street, Santa Monica, Calif
90406
ARMY ACTIVITIES
1 Chief, of Research and Development, Department of the Army
(Special Weapons and Air Defense Division), Wash, DC 20310
1 Commanding Officer, US Army Combat Developments Command,
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1 Director, Ballistic Research Laboratories (Library), Aberdeen
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1 US Army Research Office, ATTN: Lt Gregory D. Atmore, Box
CM, Duke Station, Durham, NC
1 Hq US Army Air Defense Command (ADGCB), Ent AFB, Colo
80912
1 President, US Army Air Defense Board, Ft Bliss, Tex 79916
2 Chief of Engineers (ENGMC-EM), Department of the Army,
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1 Director, Army Re search Office, 3045 Columbia Pike,
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4 Director, US Army Waterways Experiment Sta(WESRL), P. O.
Box 631, Vicksburg, Miss 39181
2 Director, US Army Engineer Research and Development
Laboratories, ATTN: STINFO Branch, Ft Belvoir) Va
NAVY ACTIVITIES
1 Chief of Naval Research, Department of the Navy, Wash, DC
20390
1 Chief, Bureau of Naval Weapons, RRNU, Department of the Navy,
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2 Bureau of Yards and Docks, Department of the Navy, Code 22. 102,
( Branch M anager, Code 42.220), Wash 25, DC
1 Commanding Officer, Naval Research Laboratory, Wash, DC
20390
1 Superintendent, US Naval Postgraduate School, ATTN: George R.
Luckett, Monterey, Calif
4 Commanding Officer and Director, Naval Civil Engineering
Laboratory, Port Hueneme, Calif
C - 3
RTD TDR-63-3096, Vol III
No. cvs
1
1
2
1
1
1
1
1
20
2
1
1
1
2
1
1
10
DISTRIBUTION (cont'd)
Officer-in-Charge, Naval Civil Engineering Corps Officer School,
US Naval Construction Battalion Center, Port Hueneme, Calif
Office of Naval Research, Wash 25, DC
OTHER DOD ACTIVITIES
Director, Defense Atomic Support Agency (Document Library
Branch), Wash, DC 20301
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Director, Weapon Systems Evaluation Group, Room ID-847, The
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Director, Advanced Research Projects Agency, Department of
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Office of Director of Defense Research and Engineering, ATTN:
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AEC ACTIVITIES
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OTHER
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20301
OTS, Department of Commerce, Wash 25, DC
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Station, Albuquerque, NM
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Johnson, 7501 N. Natchez Ave, Niles, Ill
C-4
RTD TDR- 63-3096, Vol III
No. cvs
1
1
1
1
1
1
1
1
1
1
1
1
1
1
I
1
1
1
1
20
DISTRIBUTION (cont'd)
Stanford Research Institute, 333 Ravens Wood, Menlo Park, Calif
(Mr. Ernie Chilton)
(Mr. F. N. Sauer)
(Mr. G. R. Fowles)
Portland Cement Association, Structural Development Section, ATTN
E ivind Hogne stad, 33 W. Grand Ave, Chicago, Ill
Agbabian- Jacobsen & Associates, ATTN: Drs. M. S. Agbabian and
Lydik S. Jacobsen, 8939 S. Sepulveda Blvd, Los Angeles 45, Calif
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(Mr. John Foss)
(Mr. Robert Crawford)
Allied Research Associates, ATTN: Mr. David C. Knodel, 43
Leon St, Boston, Mass
MITRON Research and Development Corp, ATTN: Dr. Maurice
Gertel, 899 Main St, Waltham, Mass
Barry Controls, Inc., ATTN: Mr. Richard Cavanaugh, 1400
Flower St, Glendale, Calif
Southwest Research Institute, ATTN: Mr. Gale Nevill, 8500
Culebra Road, San Antonio 6, Tex
Paul Weidlinger Associates, ATTN: Mr. Paul Weidlinger, 777
Third Avenue, New York, NY 10017
Shannon and Wilson, ATTN: Mr. Stanley D. Wilson, 1105 N. 38th
St, Seattle 3, Wash
The MITRE Corp, ATTN: Mr. Warren McCabe, P. O. Box 208,
Bedford, Mass
Space Technology Labs, Inc., Engineering Mechanics Dept, ATTN:
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Northrop- Ventura Corp, ATTN: Dr. J. G. Trulio, 1515 Rancho
Conejo Blvd, Newbury Park, Calif
Physics International Co., ATTN: Dr. C. S. Godfrey, 2229 Fourth
Street, Berkeley 10, Calif
West Virginia University, Dept of Civil Engineering, ATTN: Dr.
J. H. Schaub, Morgantown, WVa
North Carolina State University, Dept of Civil Engineering, ATTN:
Dr. R. E. Fadum, Raleigh, NC
University of Illinois, ATTN: Dr. Nathan M. Newmark, 207 Talbot
Laboratory, Urbana, Ill
C-5
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No. cvs
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1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
DISTRIBUTION (cont'd)
California Institute of Technology, Dept of Engineering, Pasadena,
Calif
(Prof. C. E. Crede)
(Dr. Seed)
University of Florida, Dept of Civil Engineering, ATTN: Mr. Frank
Richardt, Gainesville, Fla
Colorado School of Mines, ATTN: Mr. Dave C. Card, Golden, Colo
Grumman Aircraft Engineering Corp, ATTN: Dr. Hyman R. Garnet*
Bethpage, NY
South Dakota School of Mines and Technology, ATTN: Mr. Edwin H.
O shier, Rapid City, SD
United Electrodynamic s, Inc., ATTN: Mr. Ted Winston, 200
Allendale Road, Pasadena, Calif
Iowa State University, Dept of Theoretical and Applied Mechanics,
ATTN: Mr. Glen Murphy, Ames, la
Princeton University, Dept of Civil Engineering, Princeton, NJ
IIT Research Institute, 3422 S. Dearborn St, Chicago 15, Ill
(Dr. Eugene Seven)
(Dr. Eben Vey)
(Dr. Charles Miller)
(Dr. T. H. Schiffman)
Massachusetts Institute of Technology, Dept of Civil and Sanitary
Engineering, ATTN: Dr. Robert V. Whitman, 77 Massachusetts
Ave, Cambridge 39, Mass
Massachusetts Institute of Technology, Lincoln Laboratory
Document Library) , P. O. Box 73, Lexington, Mass 02173
University of Notre Dame, Dept of Civil Engineering, ATTN: Dr.
Harry Saxe, Notre Dame, Ind
Purdue University, Civil Engineering Dept, ATTN: Prof. G. A.
Leonards, Lafayette, Ind
Lockheed Missiles and Space Co., Technical Information Center,
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Livermore Radiation Laboratory, Plowshare Div, L 43, ATTN:
Capt Lewis Cauthen, P. O. Box 808, Livermore, Calif
The Boeing Co. , ATTN: Mr. Ron Carle son, Suite 802, First
National Bank Bldg, Albuquerque, New Mexico
C-6
RTD TDR-63-3096, Vol III
No. cvs
1
1
1
1
1
1
1
1
1
1
1 .
DISTRIBUTION (cont'd)
St Louis University, Institute of Technology, ATTN: Dr. Carl
Kisslinger, 36 21 Olive St, St Louis 8, Mo
University of Michigan, Dept of Civil Engineering, ATTN: Mr
Frank E. Richardt, Ann Arbor, Mich
University of California, College of Engineering, ATTN: Prof.
Martin Duke, Los Angeles, Calif
University of Washington, ATTN: Dr.' I. M. Fyfe, Seattle 5, Wash
Massachusetts Institute of Technology, ATTN: Prof J. P. DenHartog,
Cambridge 39, Mass
Westinghouse Research Laboratory, ATTN: Dr. E. G. Fischer,
Pittsburgh, Pa
Pennsylvania State University, ATTN: Dr. Snowden, State College, Pa
Sandia Corporation, Underground Physics Div, ATTN: Mr. Luke
J. Vortman, Sandia Base, NM 87115
National Academy of Sciences, Advisory Committee on Civil Defense,
ATTN: Mr. Richard Parks, 2101 Constitution Ave NW, Wash, DC
20418
Mechanics Research, Inc., ATTN: Dr. Robert H. Anderson,
540 Aero Space Center, 650 N. Sepulveda Blvd, El Segundo, Calif
Official Record Copy (Lt J. F. Flory, WLRC)
C-7
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